We construct a topological spin liquid (TSL) model on the kagome lattice, with SU(3) symmetry with the fundamental representation at each lattice site, based on Projected Entangled Pair States (PEPS). Using the PEPS framework, we can adiabatically connect the model to a fixed point model (analogous to the dimer model for Resonating Valence Bond states) which we prove to be locally equivalent to a $Z_3$ quantum double model. Numerical study of the interpolation reveals no sign of a phase transition or long-range order, characterizing the model conclusively as a gapped TSL. We further study the entanglement spectrum of the model and find that while it is gapped, it exhibits branches with vastly different velocities, with the slow branch matching the counting of a chiral $SU(3)_1$ CFT, suggesting that it can be deformed to a model with chiral $SU(3)_1$ entanglement spectrum.
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