  We propose and study an SU(3) trimer resonating-valence-bond (tRVB) state
with $C_{4v}$ point-group symmetry on the square lattice. By devising a
projected entangled-pair state representation, we show that all (connected)
correlation functions between local operators in this SU(3) tRVB state decay
exponentially, indicating its gapped nature. We further calculate the modular
$S$ and $T$ matrices by constructing all nine topological sectors on a torus
and establish the existence of $\mathbb{Z}_3$ topological order in this SU(3)
tRVB state.
