Not really. Lisp is a functional programming language and has existed since at least 1960. Some claim there were many other proto-functional languages since the early 60's, and the FP language [1] (a clearly functional programming language and the result of the famous paper "Can Programming Be Liberated From the von Neumann Style?") appeared in 1977 - was inspired by much earlier efforts like APL.
OOP really only became a thing with Simula in 1967, but was not popular until the 1980's with Smalltalk and Common Lisp's Object System (CLOS) came about (so yes, there was a OOP/FP hybrid already decades ago), and then C++ and finally Java much later... at which time Functional Programming languages already included Miranda (1985) which later evolved into Haskell, and Erlang (1986). That is, FPP languages were at least as common as OOP languages by the 80's.
As far as I know, however, pure functional languages were not really very efficient until Haskell came about, while OOP languages were nearly on par with procedural style: which mattered a lot in 1980's machines.
*sight*
Okay, let's dot the i's and cross the t's then, shall we, and include languages used long long before software development became the industry behemoth it is today.
You're right. There, I said it. You are absolutely right.
Many of the concepts of functional programming were indeed pioneered in LISP, and it was indeed specified in 1960, 7 years before Simula.
But hey, let's dot another "i" and correct my statement. Because, FP isn't in contrast to OOP so much as it is in contrast to imperative programming, and all it's mutable state, right?
So, is there an imperative programming language, older than LISP, and still in use today? And the answer is: Of course there is ;-)
No it isn't, because the history relevant for this question doesn't begin at the very start of programming. Most programmers active today didn't go to university in the 60s. In the timeframe relevant to answer this question, the people were indeed confronted with OOP languages, and FP languages were already a niche topic, end of story.
This also coincides with programming, as an activity, massively gaining in popularity across many industries.
Real FP means referential transparency and that started with languages like Miranda and later Haskell.
What many people consider "FP" today is in fact just procedural programming with higher order procedures and lexical closure.
A lot of early FP teaching was done in Lisp. Some kind of "Pure Lisp" was used, which are imaginary subsets of Lisp, restricted to side-effect-free, non-destructive functions.
FP started quite a bit before Miranda.
When you write Lisp, you're mostly composing functions. It's truly very functional, just not purely functional which is probably the motivation behind your refusal to include Lisp in the functional programming language family, which would horrify the founders of the field as it's a simple attempt at redefining a widely used, well understood term to become a small subset of it for reasons of gatekeeping what we should, according to you, include under the term without any real technical reason behind it.
Similar to OOP, which promised to do this by encapsulating state, FP promised to do this via purity, aka. getting rid of as much state as possible, and only allowing stateful transition at certain well defined sections of the program.
The "market advantage" of OOP was that, via Java, it was already so well established, and so many coders had been trained in OOP languages, that it remained alive. FP on the other hand, coming out of academia and requiring all these industry people to suddenly do things in syntactically and conceptually different ways, never really gained traction. OOP simply came first, it is as simple as that.
Whether FP would have actually solved the problem is anyones guess, since it never gained the traction of OOP and Procedural languages. My best guess is that it wouldn't, because I don't believe in silver bullets.
It should be noted that both approaches contributed valueable things to contemporary languages. E.g. first order functions being the norm comes from FP.