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 $\rho$.

(Asked first on MSE: https://math.stackexchange.com/questions/3096704/failure-of-schurs-lemma-for-topological-group-representations)

**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.

finite dimensjonalrepresentations, so what Nate points out is highly relevant here. $\endgroup$ – Jim Humphreys Feb 11 at 22:40