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: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=imagefirstpage_1&handle=euclid.pjm/1102690181
-For principal series representations and Steinberg: Casselman - Restriction to $GL_2(o)$: http://link.springer.com/article/10.1007%2FBF01355984#page-1 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 classifcation 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.