Lisp does not equal lambda calculus

Lisp isn't lambda calculus for the main reason that lambda calculus doesn't have conses, lists, or symbols. It cannot quote a part of itself such that the quoted part becomes data operated on by lambda calculus.

Lisp is useful even when it wrongly implements lambda calculus, with all bindings being dynamically scoped so that it only relates to lambda calculus via the naming of the lambda symbol.

But if you forbid setq and setf it is. Just replace assignments with bindings