  We report a $\mu$SR investigation of a non-centrosymmetric superconductor
(LaNiC$_2$) in single crystal form. Compared to previous $\mu$SR studies of
non-centrosymmetric superconducting polycrystalline and powder samples, the
unambiguous orientation of single crystals enables a simultaneous determination
of the absolute value of the magnetic penetration depth and the vortex core
size from measurements that probe the magnetic field distribution in the vortex
state. The magnetic field dependence of these quantities unambiguously
demonstrates the presence of two nodeless superconducting energy gaps. In
addition, we detect weak internal magnetic fields in the superconducting phase,
confirming earlier $\mu$SR evidence for a time-reversal symmetry breaking
superconducting state. Our results suggest that Cooper pairing in LaNiC$_2$ is
characterized by the same interorbital equal-spin pairing model introduced to
describe the pairing state in the centrosymmetric superconductor LaNiGa$_2$.
