Show that $GL(n , \mathbb{Z}_{p})$ has no finite dimensional faithful representation

Question

How do I show that $G = GL(n , \mathbb{Z}_{p})$ has no finite dimensional faithful representation. $\mathbb{Z}_{p}$ is the ring of $p$-adic integers.

If $\rho : G \rightarrow GL(m, \mathbb{C})$ is a finite dimensional representation then $\ker(\rho)$ is open. Can I use this fact in some way?

Is $\{1\}$ an open set in $G$?