diff --git a/frontends/p4/typeChecking/typeChecker.cpp b/frontends/p4/typeChecking/typeChecker.cpp
index 876c5a9b80..5a420e1d7d 100644
--- a/frontends/p4/typeChecking/typeChecker.cpp
+++ b/frontends/p4/typeChecking/typeChecker.cpp
@@ -3862,10 +3862,19 @@ const IR::Node *TypeInference::postorder(IR::MethodCallExpression *expression) {
             return expression;
         }
 
-        if (mi->is<ExternFunction>() && constArgs) {
-            // extern functions with constant args are compile-time constants
-            setCompileTimeConstant(expression);
-            setCompileTimeConstant(getOriginal<IR::Expression>());
+        if (const auto *ef = mi->to<ExternFunction>()) {
+            const IR::Type *baseReturnType = returnType;
+            if (const auto *sc = returnType->to<IR::Type_SpecializedCanonical>())
+                baseReturnType = sc->baseType;
+            const bool factoryOrStaticAssert =
+                baseReturnType->is<IR::Type_Extern>() || ef->method->name == "static_assert";
+            if (constArgs && factoryOrStaticAssert) {
+                // factory extern function calls (those that return extern objects) with constant
+                // args are compile-time constants.
+                // The result of a static_assert call is also a compile-time constant.
+                setCompileTimeConstant(expression);
+                setCompileTimeConstant(getOriginal<IR::Expression>());
+            }
         }
 
         auto bi = mi->to<BuiltInMethod>();
diff --git a/testdata/p4_16_samples/issue4661_non_pure_extern_function_const_args.p4 b/testdata/p4_16_samples/issue4661_non_pure_extern_function_const_args.p4
new file mode 100644
index 0000000000..3863a9f07f
--- /dev/null
+++ b/testdata/p4_16_samples/issue4661_non_pure_extern_function_const_args.p4
@@ -0,0 +1,24 @@
+#include <core.p4>
+
+extern void foo();
+extern void bar();
+extern bit<8> baz();
+action a(){}
+action b(){}
+control c() {
+    table t  {
+        actions = { a ; b;  }
+    }
+    apply {
+        switch(baz()) {
+            1 : { foo(); }
+            4 : { bar(); }
+        }
+        t.apply();
+    }
+}
+
+control C();
+package top(C c);
+
+top(c()) main;
diff --git a/testdata/p4_16_samples/issue4661_pure_extern_function_const_args.p4 b/testdata/p4_16_samples/issue4661_pure_extern_function_const_args.p4
new file mode 100644
index 0000000000..a39c23a3ec
--- /dev/null
+++ b/testdata/p4_16_samples/issue4661_pure_extern_function_const_args.p4
@@ -0,0 +1,24 @@
+#include <core.p4>
+
+extern void foo();
+extern void bar();
+@pure extern bit<8> baz();
+action a(){}
+action b(){}
+control c() {
+    table t  {
+        actions = { a ; b;  }
+    }
+    apply {
+        switch(baz()) {
+            1 : { foo(); }
+            4 : { bar(); }
+        }
+        t.apply();
+    }
+}
+
+control C();
+package top(C c);
+
+top(c()) main;
diff --git a/testdata/p4_16_samples_outputs/issue4656_const_fold_generic_switch_label_expr.p4-stderr b/testdata/p4_16_samples_outputs/issue4656_const_fold_generic_switch_label_expr.p4-stderr
index 72a9bd7b51..e69de29bb2 100644
--- a/testdata/p4_16_samples_outputs/issue4656_const_fold_generic_switch_label_expr.p4-stderr
+++ b/testdata/p4_16_samples_outputs/issue4656_const_fold_generic_switch_label_expr.p4-stderr
@@ -1,3 +0,0 @@
-issue4656_const_fold_generic_switch_label_expr.p4(13): [--Wwarn=mismatch] warning: baz(): constant expression in switch
-        switch(baz()) {
-               ^^^^^
diff --git a/testdata/p4_16_samples_outputs/issue4661_non_pure_extern_function_const_args-first.p4 b/testdata/p4_16_samples_outputs/issue4661_non_pure_extern_function_const_args-first.p4
new file mode 100644
index 0000000000..64c764fcc6
--- /dev/null
+++ b/testdata/p4_16_samples_outputs/issue4661_non_pure_extern_function_const_args-first.p4
@@ -0,0 +1,34 @@
+#include <core.p4>
+
+extern void foo();
+extern void bar();
+extern bit<8> baz();
+action a() {
+}
+action b() {
+}
+control c() {
+    table t {
+        actions = {
+            a();
+            b();
+            @defaultonly NoAction();
+        }
+        default_action = NoAction();
+    }
+    apply {
+        switch (baz()) {
+            8w1: {
+                foo();
+            }
+            8w4: {
+                bar();
+            }
+        }
+        t.apply();
+    }
+}
+
+control C();
+package top(C c);
+top(c()) main;
diff --git a/testdata/p4_16_samples_outputs/issue4661_non_pure_extern_function_const_args-frontend.p4 b/testdata/p4_16_samples_outputs/issue4661_non_pure_extern_function_const_args-frontend.p4
new file mode 100644
index 0000000000..f7f9760cb6
--- /dev/null
+++ b/testdata/p4_16_samples_outputs/issue4661_non_pure_extern_function_const_args-frontend.p4
@@ -0,0 +1,38 @@
+#include <core.p4>
+
+extern void foo();
+extern void bar();
+extern bit<8> baz();
+control c() {
+    @name("c.tmp") bit<8> tmp;
+    @name(".a") action a_0() {
+    }
+    @name(".b") action b_0() {
+    }
+    @noWarn("unused") @name(".NoAction") action NoAction_1() {
+    }
+    @name("c.t") table t_0 {
+        actions = {
+            a_0();
+            b_0();
+            @defaultonly NoAction_1();
+        }
+        default_action = NoAction_1();
+    }
+    apply {
+        tmp = baz();
+        switch (tmp) {
+            8w1: {
+                foo();
+            }
+            8w4: {
+                bar();
+            }
+        }
+        t_0.apply();
+    }
+}
+
+control C();
+package top(C c);
+top(c()) main;
diff --git a/testdata/p4_16_samples_outputs/issue4661_non_pure_extern_function_const_args-midend.p4 b/testdata/p4_16_samples_outputs/issue4661_non_pure_extern_function_const_args-midend.p4
new file mode 100644
index 0000000000..386a146678
--- /dev/null
+++ b/testdata/p4_16_samples_outputs/issue4661_non_pure_extern_function_const_args-midend.p4
@@ -0,0 +1,88 @@
+#include <core.p4>
+
+extern void foo();
+extern void bar();
+extern bit<8> baz();
+control c() {
+    @name("c.tmp") bit<8> tmp;
+    @name(".a") action a_0() {
+    }
+    @name(".b") action b_0() {
+    }
+    @noWarn("unused") @name(".NoAction") action NoAction_1() {
+    }
+    @name("c.t") table t_0 {
+        actions = {
+            a_0();
+            b_0();
+            @defaultonly NoAction_1();
+        }
+        default_action = NoAction_1();
+    }
+    @hidden action switch_0_case() {
+    }
+    @hidden action switch_0_case_0() {
+    }
+    @hidden action switch_0_case_1() {
+    }
+    @hidden table switch_0_table {
+        key = {
+            tmp: exact;
+        }
+        actions = {
+            switch_0_case();
+            switch_0_case_0();
+            switch_0_case_1();
+        }
+        const default_action = switch_0_case_1();
+        const entries = {
+                        const 8w1 : switch_0_case();
+                        const 8w4 : switch_0_case_0();
+        }
+    }
+    @hidden action issue4661_non_pure_extern_function_const_args14() {
+        foo();
+    }
+    @hidden action issue4661_non_pure_extern_function_const_args15() {
+        bar();
+    }
+    @hidden action act() {
+        tmp = baz();
+    }
+    @hidden table tbl_act {
+        actions = {
+            act();
+        }
+        const default_action = act();
+    }
+    @hidden table tbl_issue4661_non_pure_extern_function_const_args14 {
+        actions = {
+            issue4661_non_pure_extern_function_const_args14();
+        }
+        const default_action = issue4661_non_pure_extern_function_const_args14();
+    }
+    @hidden table tbl_issue4661_non_pure_extern_function_const_args15 {
+        actions = {
+            issue4661_non_pure_extern_function_const_args15();
+        }
+        const default_action = issue4661_non_pure_extern_function_const_args15();
+    }
+    apply {
+        tbl_act.apply();
+        switch (switch_0_table.apply().action_run) {
+            switch_0_case: {
+                tbl_issue4661_non_pure_extern_function_const_args14.apply();
+            }
+            switch_0_case_0: {
+                tbl_issue4661_non_pure_extern_function_const_args15.apply();
+            }
+            switch_0_case_1: {
+            }
+        }
+        t_0.apply();
+    }
+}
+
+control C();
+package top(C c);
+top(c()) main;
diff --git a/testdata/p4_16_samples_outputs/issue4661_non_pure_extern_function_const_args.p4 b/testdata/p4_16_samples_outputs/issue4661_non_pure_extern_function_const_args.p4
new file mode 100644
index 0000000000..1343b2fc19
--- /dev/null
+++ b/testdata/p4_16_samples_outputs/issue4661_non_pure_extern_function_const_args.p4
@@ -0,0 +1,32 @@
+#include <core.p4>
+
+extern void foo();
+extern void bar();
+extern bit<8> baz();
+action a() {
+}
+action b() {
+}
+control c() {
+    table t {
+        actions = {
+            a;
+            b;
+        }
+    }
+    apply {
+        switch (baz()) {
+            1: {
+                foo();
+            }
+            4: {
+                bar();
+            }
+        }
+        t.apply();
+    }
+}
+
+control C();
+package top(C c);
+top(c()) main;
diff --git a/testdata/p4_16_samples_outputs/issue4661_non_pure_extern_function_const_args.p4-stderr b/testdata/p4_16_samples_outputs/issue4661_non_pure_extern_function_const_args.p4-stderr
new file mode 100644
index 0000000000..e69de29bb2
diff --git a/testdata/p4_16_samples_outputs/issue4661_pure_extern_function_const_args-first.p4 b/testdata/p4_16_samples_outputs/issue4661_pure_extern_function_const_args-first.p4
new file mode 100644
index 0000000000..bcb1350467
--- /dev/null
+++ b/testdata/p4_16_samples_outputs/issue4661_pure_extern_function_const_args-first.p4
@@ -0,0 +1,34 @@
+#include <core.p4>
+
+extern void foo();
+extern void bar();
+@pure extern bit<8> baz();
+action a() {
+}
+action b() {
+}
+control c() {
+    table t {
+        actions = {
+            a();
+            b();
+            @defaultonly NoAction();
+        }
+        default_action = NoAction();
+    }
+    apply {
+        switch (baz()) {
+            8w1: {
+                foo();
+            }
+            8w4: {
+                bar();
+            }
+        }
+        t.apply();
+    }
+}
+
+control C();
+package top(C c);
+top(c()) main;
diff --git a/testdata/p4_16_samples_outputs/issue4661_pure_extern_function_const_args-frontend.p4 b/testdata/p4_16_samples_outputs/issue4661_pure_extern_function_const_args-frontend.p4
new file mode 100644
index 0000000000..531c26a86a
--- /dev/null
+++ b/testdata/p4_16_samples_outputs/issue4661_pure_extern_function_const_args-frontend.p4
@@ -0,0 +1,38 @@
+#include <core.p4>
+
+extern void foo();
+extern void bar();
+@pure extern bit<8> baz();
+control c() {
+    @name("c.tmp") bit<8> tmp;
+    @name(".a") action a_0() {
+    }
+    @name(".b") action b_0() {
+    }
+    @noWarn("unused") @name(".NoAction") action NoAction_1() {
+    }
+    @name("c.t") table t_0 {
+        actions = {
+            a_0();
+            b_0();
+            @defaultonly NoAction_1();
+        }
+        default_action = NoAction_1();
+    }
+    apply {
+        tmp = baz();
+        switch (tmp) {
+            8w1: {
+                foo();
+            }
+            8w4: {
+                bar();
+            }
+        }
+        t_0.apply();
+    }
+}
+
+control C();
+package top(C c);
+top(c()) main;
diff --git a/testdata/p4_16_samples_outputs/issue4661_pure_extern_function_const_args-midend.p4 b/testdata/p4_16_samples_outputs/issue4661_pure_extern_function_const_args-midend.p4
new file mode 100644
index 0000000000..4f080dde33
--- /dev/null
+++ b/testdata/p4_16_samples_outputs/issue4661_pure_extern_function_const_args-midend.p4
@@ -0,0 +1,88 @@
+#include <core.p4>
+
+extern void foo();
+extern void bar();
+@pure extern bit<8> baz();
+control c() {
+    @name("c.tmp") bit<8> tmp;
+    @name(".a") action a_0() {
+    }
+    @name(".b") action b_0() {
+    }
+    @noWarn("unused") @name(".NoAction") action NoAction_1() {
+    }
+    @name("c.t") table t_0 {
+        actions = {
+            a_0();
+            b_0();
+            @defaultonly NoAction_1();
+        }
+        default_action = NoAction_1();
+    }
+    @hidden action switch_0_case() {
+    }
+    @hidden action switch_0_case_0() {
+    }
+    @hidden action switch_0_case_1() {
+    }
+    @hidden table switch_0_table {
+        key = {
+            tmp: exact;
+        }
+        actions = {
+            switch_0_case();
+            switch_0_case_0();
+            switch_0_case_1();
+        }
+        const default_action = switch_0_case_1();
+        const entries = {
+                        const 8w1 : switch_0_case();
+                        const 8w4 : switch_0_case_0();
+        }
+    }
+    @hidden action issue4661_pure_extern_function_const_args14() {
+        foo();
+    }
+    @hidden action issue4661_pure_extern_function_const_args15() {
+        bar();
+    }
+    @hidden action act() {
+        tmp = baz();
+    }
+    @hidden table tbl_act {
+        actions = {
+            act();
+        }
+        const default_action = act();
+    }
+    @hidden table tbl_issue4661_pure_extern_function_const_args14 {
+        actions = {
+            issue4661_pure_extern_function_const_args14();
+        }
+        const default_action = issue4661_pure_extern_function_const_args14();
+    }
+    @hidden table tbl_issue4661_pure_extern_function_const_args15 {
+        actions = {
+            issue4661_pure_extern_function_const_args15();
+        }
+        const default_action = issue4661_pure_extern_function_const_args15();
+    }
+    apply {
+        tbl_act.apply();
+        switch (switch_0_table.apply().action_run) {
+            switch_0_case: {
+                tbl_issue4661_pure_extern_function_const_args14.apply();
+            }
+            switch_0_case_0: {
+                tbl_issue4661_pure_extern_function_const_args15.apply();
+            }
+            switch_0_case_1: {
+            }
+        }
+        t_0.apply();
+    }
+}
+
+control C();
+package top(C c);
+top(c()) main;
diff --git a/testdata/p4_16_samples_outputs/issue4661_pure_extern_function_const_args.p4 b/testdata/p4_16_samples_outputs/issue4661_pure_extern_function_const_args.p4
new file mode 100644
index 0000000000..86fe060301
--- /dev/null
+++ b/testdata/p4_16_samples_outputs/issue4661_pure_extern_function_const_args.p4
@@ -0,0 +1,32 @@
+#include <core.p4>
+
+extern void foo();
+extern void bar();
+@pure extern bit<8> baz();
+action a() {
+}
+action b() {
+}
+control c() {
+    table t {
+        actions = {
+            a;
+            b;
+        }
+    }
+    apply {
+        switch (baz()) {
+            1: {
+                foo();
+            }
+            4: {
+                bar();
+            }
+        }
+        t.apply();
+    }
+}
+
+control C();
+package top(C c);
+top(c()) main;
diff --git a/testdata/p4_16_samples_outputs/issue4661_pure_extern_function_const_args.p4-stderr b/testdata/p4_16_samples_outputs/issue4661_pure_extern_function_const_args.p4-stderr
new file mode 100644
index 0000000000..e69de29bb2