There is no such thing as a non-cons-cell-based Lisp.
Just like there is no non-H2O water.
Well, there are isotopes: D2O is "heavy water". Similarly, not all Lisps have exactly the same cons representation down to the sub-atomic level; there are various isotopes.
To my mind the defining characteristic of a Lisp is that code is represented as a first-class data structure, one that can be produced by parsing text, but that has more structure than text. In the case of most Lisp's that first-class structure is a cons cell, and I think that is by far the best choice, but I don't think that using a different structure should be disqualifying by definition. If you're really going to be that strict about it, then Scheme is not a Lisp because its standard does not actually specify the data structure that is used. It only specifies that code can be represented "as data" but it doesn't say what kind of data that is. It could be a string and still conform to the standard.
Which Lisp code are you asserting it can not run? Scheme? CommonLisp? Racket? Emacs Lisp?
Furthermore which of those Lisp dialects can or cannot run each other’s code? Is your definition of “Lisp code” a primordial core of which a set of languages descend? If so, what is that primordial core?
