  We analyze the Kitaev model on the $\{9,3\}$ hyperbolic lattice. The
$\{9,3\}$ is formed by a regular 3-coordinated tiling of nonagons, where the
3-color coding of bonds according to the inequivalent Kitaev Ising spin
couplings yields the natural generalization of the original Kitaev model for
Euclidean regular honeycomb tiling. Upon investigation of the bulk spectrum for
large finite size droplets, we identify a gapless chiral $\mathbb{Z}_2$ spin
liquid state featuring spontaneous time reversal symmetry breaking. Due to its
non-commutative translation group structure, such type of hyperbolic spin
liquid is conjectured to feature chiral quasiparticles with a potentially
non-Abelian Bloch profile.
