Sigh. Immutability is a relatively recent addition to the long list of features that defines a language as "functional."
In the beginning, "functional" meant functions were first-class objects and there was little--if any--global state. Lisp passes this test. Lisp invented this test (especially Scheme which introduced lexical scoping and closures).
If you want to write code with immutable data, you can certainly do so with Lisp; it just doesn't force you to. For the most part Lisp creates a brand new data structure when you "modify" an old one. You have to work a bit harder to actually mutate things.
Where e.g. Haskell is "more" functional than Lisp is more about automatic currying than immutability. Plus the static type system that allows monadic programming, which is much more difficult in Lisp because of Lisp's dynamic typing.