  We study a $\mathbb{Z}_3$ Kitaev model on the honeycomb lattice with nearest
neighbor interactions. Based on matrix product state simulations and symmetry
considerations, we find evidence that, with ferromagnetic isotropic couplings,
the model realizes a chiral spin liquid, characterized by a possible
$\mathrm{U}(1)_{12}$ chiral topological order. This is supported by simulations
on both cylinder and strip geometries. On infinitely long cylinders with
various widths, scaling analysis of entanglement entropy and maximal
correlation length suggests that the model has a gapped 2D bulk. The
topological entanglement entropy is extracted and found to be in agreement with
the $\mathrm{U}(1)_{12}$ topological order. On infinitely long strips with
moderate widths, we find the model is critical with a central charge consistent
with the chiral edge theory of the $\mathrm{U}(1)_{12}$ topological phase. We
conclude by discussing several open questions.
