Most Lisps are multiparadigm, covering at least the functional and procedural paradigms. Going to Common Lisp you can add OO to the mix. It is a severely constrained notion of FP that manages to exclude the first functional programming language from its definition.
It's also worth mentioning that the Common Lisp standard does not require implementations to implement tail call elimination. The pervasive use of a functional style in CL would therefore give rise to programs with performance characteristics that could, at least in principle, vary greatly between implementations. (I'm aware that most popular CL implementations do optimize away tail calls in practice.)
Common Lisp doesn't require implementations to provide tail call support, so functional programs that express iteration using recursion may be severely limited in the inputs that they can handle; they would typically be ported to Common Lisp by a rewrite using iteration.