diff --git a/Lib/pickle.py b/Lib/pickle.py
--- a/Lib/pickle.py
+++ b/Lib/pickle.py
@@ -476,6 +476,11 @@ class _Pickler:
             f(self, obj) # Call unbound method with explicit self
             return
 
+        self.save_user_defined(obj)
+
+    def save_user_defined(self, obj):
+        # Save object with object-defined methods (reduce and copyreg)
+        t = type(obj)
         # Check private dispatch table if any, or else copyreg.dispatch_table
         reduce = getattr(self, 'dispatch_table', dispatch_table).get(t)
         if reduce is not None:
@@ -973,7 +978,16 @@ class _Pickler:
             return self.save_reduce(type, (...,), obj=obj)
         return self.save_global(obj)
 
-    dispatch[FunctionType] = save_global
+    def save_function(self, obj):
+        try:
+            return self.save_global(obj)
+        except PicklingError:
+            # fall back to reduce and clear the exception
+            pass
+
+        return self.save_user_defined(obj)
+
+    dispatch[FunctionType] = save_function
     dispatch[type] = save_type
 
 
diff --git a/Lib/test/pickletester.py b/Lib/test/pickletester.py
--- a/Lib/test/pickletester.py
+++ b/Lib/test/pickletester.py
@@ -657,6 +657,13 @@ def create_data():
     return x
 
 
+def unpickle_lambda(name, argdefs, *co_args):
+    code_type = type(unpickle_lambda.__code__)
+    code = code_type(*co_args)
+    fun_type = type(unpickle_lambda)
+    return fun_type(code, {}, name, argdefs)
+
+
 class AbstractUnpickleTests(unittest.TestCase):
     # Subclass must define self.loads.
 
@@ -2259,20 +2266,65 @@ class AbstractPickleTests(unittest.TestC
             pass
         # Since the function is local, lookup will fail
         for proto in range(0, pickle.HIGHEST_PROTOCOL + 1):
-            with self.assertRaises((AttributeError, pickle.PicklingError)):
+            with self.assertRaises((AttributeError, TypeError, pickle.PicklingError)):
                 pickletools.dis(self.dumps(f, proto))
         # Same without a __module__ attribute (exercises a different path
         # in _pickle.c).
         del f.__module__
         for proto in range(0, pickle.HIGHEST_PROTOCOL + 1):
-            with self.assertRaises((AttributeError, pickle.PicklingError)):
+            with self.assertRaises((AttributeError, TypeError, pickle.PicklingError)):
                 pickletools.dis(self.dumps(f, proto))
         # Yet a different path.
         f.__name__ = f.__qualname__
         for proto in range(0, pickle.HIGHEST_PROTOCOL + 1):
-            with self.assertRaises((AttributeError, pickle.PicklingError)):
+            with self.assertRaises((AttributeError, TypeError, pickle.PicklingError)):
                 pickletools.dis(self.dumps(f, proto))
 
+    def test_custom_functions_pickle(self):
+        f = lambda: 42
+
+        def pickle_lambda(fun):
+            # Simplified pickling for testing purposes:
+            # only pure lambdas, even no imports handled
+            name = fun.__name__
+            if name != '<lambda>':
+                raise pickle.PicklingError(
+                    "Can't pickle function object: %s" % (name,))
+
+            if fun.__closure__ is not None:
+                raise pickle.PicklingError(
+                    "Can't pickle closures, only pure lambda allowed: %s" % (name,))
+
+            co = fun.__code__
+            args = (
+                name,
+                fun.__defaults__,
+                co.co_argcount,
+                co.co_kwonlyargcount,
+                co.co_nlocals,
+                co.co_stacksize,
+                co.co_flags,
+                co.co_code,
+                co.co_consts,
+                co.co_names,
+                co.co_varnames,
+                co.co_filename,
+                co.co_name,
+                co.co_firstlineno,
+                co.co_lnotab,
+                co.co_freevars,
+                co.co_cellvars,
+            )
+            return unpickle_lambda, args
+
+        dt = pickle.dispatch_table.copy()
+        dt[type(f)] = pickle_lambda
+        i = io.BytesIO()
+
+        pickler = self.pickler(i)
+        pickler.dispatch_table = dt
+        pickler.dump(f)
+        self.assertEqual(f(), self.loads(i.getvalue())())
 
 class BigmemPickleTests(unittest.TestCase):
 
diff --git a/Modules/_pickle.c b/Modules/_pickle.c
--- a/Modules/_pickle.c
+++ b/Modules/_pickle.c
@@ -3869,8 +3869,18 @@ save(PicklerObject *self, PyObject *obj,
         goto done;
     }
     else if (type == &PyFunction_Type) {
+        PickleState *st;
         status = save_global(self, obj, NULL);
-        goto done;
+        st = _Pickle_GetGlobalState();
+        if (status < 0
+            && (PyErr_ExceptionMatches(st->PicklingError)
+                || PyErr_ExceptionMatches(PyExc_AttributeError))) {
+            /* fall back to reduce */
+            PyErr_Clear();
+        }
+        else {
+            goto done;
+        }
     }
 
     /* XXX: This part needs some unit tests. */