  Material realizations of the bond-dependent Kitaev interactions with $S$=1/2
local moments have vitalized the research in quantum spin liquids. Recently, it
has been proposed that higher-spin analogues of the Kitaev interactions may
also occur in a number of materials with strong spin-orbit coupling. In
contrast to the celebrated $S$=1/2 Kitaev model on the honeycomb lattice, the
higher-spin Kitaev models are not exactly solvable. Hence, the existence of
quantum spin liquids in these systems remains an outstanding question. In this
work, we use the density matrix renormalization group (DMRG) methods to
numerically investigate the $S$=1 Kitaev model with both ferromagnetic (FM) and
antiferromagnetic (AFM) interactions. Using results on a cylindrical geometry
with various circumferences, we conclude that the ground state of the $S$=1
Kitaev model is a quantum spin liquid with a $\mathbb{Z}_2$ gauge structure. We
also put a bound on the excitation gap, which turns out to be quite small. The
magnetic field responses for the FM and AFM models are similar to those of the
$S$=1/2 counterparts. In particular, in the AFM $S$=1 model, a gapless quantum
liquid state emerges in an intermediate window of magnetic field strength,
before the system enters a trivial polarized state.
