Not really. FP has an obvious connection to lambda calculus. Many OOP languages don't even have a straightforward notion of "function", which is what lambda calculus is all about.
See e.g. Landin, P. J. (1965): A Correspondence between ALGOL 60 and Church's Lambda-notation.
> There is a well-established theory of functions, the λ-calculus, ... However, our theory of objects is self-contained; it is the first that does not require explicit reference to functions or procedures.
So that answers your question in the negative and proves my point: no, not every language is based on lambda calculus.