The usual reason people give for believing this is the inability to mutate closed-over variables.
If that's the case, then a strictly-immutable FP language similarly "wouldn't have closures" and thus could not be FP. Problem solved!
def some_function(val):
def say_stuff():
print("message is: {}".format(val))
def mutate_me(new_mutation):
nonlocal val
val = new_mutation
say_stuff.mutate_me = mutate_me
return say_stuff
f = some_function("hello")
f.mutate_me("goodbye")
f()