You mean restriction of irreducible smooth, admissible representations of $G(F)$ to $G(o)$ decomposes with multiplicity one? Then yes for $n=2$. 

Here are some more exact references:

-One-dimensional representations are obvious.

-For supercuspidal representations: Kristina Hansen. "Restriction to ${\rm GL}_2({\scr O})$ of supercuspidal representations of ${\rm GL}_2(F)$", Pacific J. Math. 130 (2) 327 - 349, 1987. https://projecteuclid.org/euclid.pjm/1102690181

-For principal series representations and Steinberg: Casselman - Restriction to $GL_2(o)$: https://doi.org/10.1007/BF01355984
Note here that $Res_{G(o)} Ind(B(F))^{G(F)} \mu = Ind_{B(o)}^{G(o)} \mu$, which Casselman gives an explicit decomposition in the first lemma. The Steinberg as a quotient/submodule of some $Ind_{B(F)}^G(F) \mu$ for some $\mu$ has then also the property.


I don't know a more conceptual proof. You need classification of all irreducible smooth admissible representations and then you need to look at the restriction, though. It's annoying. For $n>3$, we don't actually know the representation theory of $GL_n(Z_p)$, so I thing it's pretty much open. The corresponding question for $GL(n,R)$ or $GL(n, C)$ seem to be wrong for $n>3$, but are right for $n=2$. I would only care about the types necessary to classify smooth admissible representations. There I think you have a possitive answer meaning they occur with single multiplicity in irreducible admissible represenations.