Not exactly. See McCarthy's history of LISP paper.
McCarthy had implemented some of LISP's key concepts in FLPL prior to 1960. Circa 1958 Wiesner, then Director of MIT RLE, gave McCarthy and Minsky resources for the MIT Artificial Intelligence Project. Some of those were put toward programming-language development. Development began on what would become LISP in '58.
In 1960, McCarthy wrote his seminal LISP paper ("Recursive functions of symbolic expressions and their computation by machine, part I"). That "describ[ed] LISP both as a programming language and as a formalism", so even at that point it wasn't "just a theoretical" anything – it was already conceived of as a usable programming language. (And FLPL, and other list-processing languages such as ICL, supported that view.)
"Recursive functions" described the eval function and a suitable notation for expressing it, though McCarthy writes "such a notation was devised for the purposes of the paper with no thought that it would be used to express LISP programs in practice". Then he adds:
S.R. Russell noticed that eval could serve as an interpreter for LISP,
promptly hand coded it, and we now had a programming language with an
That was certainly an important moment, both for REPLs and for the future of LISP, since it ended up killing M-expressions in favor of S-expressions for LISP source.
Note that this was in 1960 on an IBM 704. Contrary to what someone posted upthread, the first interactive LISP was not the 1963 PDP-1 LISP.