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 $K$.