  We compute the axial anomaly of a Lifshitz fermion theory with anisotropic
scaling z=3 which is minimally coupled to geometry in 3+1 space-time
dimensions. We find that the result is identical to the relativistic case using
path integral methods. An independent verification is provided by showing with
spectral methods that the eta-invariant of the Dirac and Lifshitz fermion
operators in three dimensions are equal. Thus, by the integrated form of the
anomaly, the index of the Dirac operator still accounts for the possible
breakdown of chiral symmetry in non-relativistic theories of gravity. We apply
this framework to the recently constructed gravitational instanton backgrounds
of Horava-Lifshitz theory and find that the index is non-zero provided that the
space-time foliation admits leaves with harmonic spinors. Using Hitchin's
construction of harmonic spinors on Berger spheres, we obtain explicit results
for the index of the fermion operator on all such gravitational instanton
backgrounds with SU(2)xU(1) isometry. In contrast to the instantons of Einstein
gravity, chiral symmetry breaking becomes possible in the unimodular phase of
Horava-Lifshitz theory arising at lambda = 1/3 provided that the volume of
space is bounded from below by the ratio of the Ricci to Cotton tensor
couplings raised to the third power. Some other aspects of the anomalies in
non-relativistic quantum field theories are also discussed.
