Is there an example of $G$, $\rho$ as below?
$G$ is a locally compact group.
$\rho$ is an irreducible continuous representation of $G$ on a complex Hilbert space $V$. This means that we have a continuous homomorphism from $G$ to the group of bounded linear operators on $V$ with bounded inverse, such that $G \times V \rightarrow V$ is continuous. And there are no nontrivial closed invariant subspaces.
Schur's lemma fails for $\rho$.
Added later: For concreteness, I will record below one version of Schur's lemma that I have in mind. (But I'm not too picky about this.)
Schur's lemma: Every bounded operator commuting with $\rho(G)$ is a scalar.