  We study the spectrum of fermion states localized within the vortex core of a
weak-coupling p-wave superfluid. The low energy spectrum consists of two
anomalous branches that cross the Fermi level as functions of the impact
parameter, and generate large density of states at the locations of the half
cores of the vortex. Fermi liquid interactions significantly change the vortex
structure, which leads to Lifshitz transition in the effective Fermi surface of
the vortex core fermions. We apply the results to revise the interpretation of
an experiment on rotational dynamics of vortices in superfluid 3He-B.
