Let $G$ be a simple linear group over a non-archimedean local field $F$. Assume that the split-rank over $F$ is at least 2. Let $\Gamma$ be a lattice in $G$. Then $\Gamma$ is a finitely generated Kazhdan group.

My question is this: Does $\Gamma$ admit a finite-dimensional unitary representation $\rho$ such that the image $\rho(\Gamma)$ is infinite?