You mean restriction of irreducibele smooth, admissible representations of $G(F)$ to $G(o)$ decomposes single multiplicity? Then yes for $n=2$. This follows from the theory of types, but should be already in Silberger's LNM.
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
I don't know a more conceptual proof. You need classiifcation of all irreducible smooth admissible representations and then restricting, 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.