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The original LISP was influenced by Alonzo Church's lambda calculus, see: https://dl.acm.org/citation.cfm?id=367199 - however if you'll study its history the first Lisps were only experiments and for example they didn't have lexical scoping, but dynamic scoping. The Lisp descendant that made FP doable was Scheme, bringing lexical scoping and call/cc.

And today's Common Lisp is definitely not Scheme, or an FP language. You can do FP in CLisp of course, since it's quite capable, but CLisp is a general purpose language and is used as such. And in my small experience from playing with it, there isn't much FP in CLisp, but YMMV.

Lisp in general is a big family. Emacs Lisp for example has absolutely nothing to do with FP.

Of course you can romanticize about Lisp and it definitely has some cool descendants like Clojure, but you know, don't do it too much :-)