Though Peter's answer addresses the finite-dimensional representation theory, I believe that the question asks about the unitary representations on Hilbert spaces, and more general irreps on Banach and Frechet spaces.

This question has been the subject of much recent work by Avraham Aizenbud, Dmitry Gourevitch, Steve Rallis, Gerard Schiffmann, and Eitan Sayag. In particular, Aizenbud and Gourevitch prove the following in their paper "Multiplicity One Theorem for $(GL_{n+1}(R), GL_n(R))$":

Let $F = R$ or $F = C$. Let $\pi$ and $\tau$ be irreducible admissible smooth Fr\'echet representations of $GL_{n+1}(F)$ and $GL_n(F)$, respectively. Then
$$dim \left( Hom_{GL_n(F)}(\pi, \tau) \right) \leq 1.$$

This paper is on the ArXiv, and now published in Selecta, according to Aizenbud's webpage.

Zhu and Binyong have also proved this, I believe. The result has also been proven for irreducible smooth repreesentations of $GL_{n+1}(F)$ and $GL_n(F)$, when $F$ is a $p$-adic field by Aizenbud-Gourevitch-Rallis-Schiffmann.

Considering the smooth Fr\'echet case should suffice for the case of unitary representations on Hilbert spaces, I believe, by considering the subspace of smooth vectors and Garding's theorem. I'd guess it would also work for Banach space representations, but I'm not an expert on these analytic things.

It's important to note that semisimplicity may be lost when one restricts smooth representations in these settings -- so their theorem says something about occurrences of quotients after restriction. It's important to be careful about the meaning of "multiplicity-free" in these situations.