Is there an example of $G$, $\rho$ as below?
$G$ is a locally compact group.
$\rho$ is a continuous representation of $G$ on a Hilbert space $V$. This means that we have a homomorphism from $G$ to the group of bounded linear operators on $V$ with bounded inverse, such that $G \times V \rightarrow V$ is continuous.
Schur's lemma fails for $\rho$.