Lisp doesn't actually exist. Lisp is just axioms. You run an implementation of Lisp allowing various operations. Some flavors actually have (practically) zero mutability. You just create functions and cons cells and the rest of the world gets built on top of those.
But, pure functional systems are useless (to be purely functional would mean you can't even do I/O... your program would just heat up the computer, which, in a way, is its own form of output, so you can't ever be fully functional).
Every functional language gets non-functional at some point for data processing, bootstrapping, loading/saving programs, and the best functional languages even have optional mutability constructs for performance (see: !). You can't create the most performant data structures possible if all your operations require path copying, but mutability should be reserved for internal high performance library creation and avoided in business logic/application level code to make systems easier to reason about.