Say $p$ and $\ell$ are distinct prime numbers.
Let $G$ be a pro-$p$-group which acts continuously on a finite-dimensional $\mathbb{Q}_\ell$-vector space $V$.
Assume that the action of $G$ on $V$ is unipotent, i.e. $\exists n$ such that $(\sigma - 1)^n = 0$ for all $\sigma \in G$.
Does it follow that the action of $G$ on $V$ is trivial?