Let $p$ be a prime number, $m,n \in \mathbb{N}$, $F = F(p,m)$ be the free pro-$p$ group on $m$ generators. For which $(m,n)$ there is a continuous faithful representation (embedding) $\rho : F \rightarrow \text{GL}\_n(\overline{\mathbb{Q}\_p})$?


Note that every compact subgroup of $\text{GL}\_n(\overline{\mathbb{Q}\_p})$ is conjugate to a subgroup of $\text{GL}\_n(\mathcal{O}\_K)$ where $K$ is a finite extension of $\mathbb{Q}\_p$ and $\mathcal{O}\_K$ is the integral closure of $\mathbb{Z}\_p$ in $K$.



Faithful representations of free pro-p groups
It will only happen if $m=1$. See this paper:


<http://mlarsen.math.indiana.edu/~larsen/papers/2gen.pdf>


Indeed, the pro-$p$ groups that are linear over local fields of characteristic $0$ are just the pro-$p$ groups of finite subgroup rank.


Edit: As Ian Agol suggested, you should look at 'Analytic pro-p Groups' by Dixon, de Sautoy, Mann and Segal: 


books.google.com/books?id=Fjq-ngEACAAJ


It develops a remarkably powerful theory (sorry about the pun) for these groups and is well worth looking at if you are interested in linear groups over the p-adics (and extensions thereof).


