  We study the effects of quantum fluctuations on the dynamical generation of a
gap and on the evolution of the spin-wave spectra of a frustrated magnet on a
triangular lattice with bond-dependent Ising couplings, analog of the Kitaev
honeycomb model. The quantum fluctuations lift the subextensive degeneracy of
the classical ground-state manifold by a quantum order-by-disorder mechanism.
Nearest-neighbor chains remain decoupled and the surviving discrete degeneracy
of the ground state is protected by a hidden model symmetry. We show how the
four-spin interaction, emergent from the fluctuations, generates a spin gap
shifting the nodal lines of the linear spin-wave spectrum to finite energies.
