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Non-centrosymmetric superconductors (NCSs) lack inversion center in their crystal structure, and consequently one of the typical symmetries for the formation of Cooper pairs. The absence of inversion symmetry strongly influences the Cooper pairing states possible in these materials. Here we report a muon spin rotation/relaxation (μSR) study of LaNiC₂ single crystals, a non-magnetic, NCS with weak electronic correlations and a superconducting transition temperature, T_c ~ 2.7 K. Previous zero-field μSR measurements on a pressed pellet of LaNiC₂ powder revealed the onset of spontaneous magnetic fields at T_c. This finding signifies time-reversal symmetry breaking (TRSB) in superconducting phase, which is compatible with a non-unitary triplet-pairing state. However, other kinds of experiments have provided evidence for conventional BCS s-wave pairing, point nodes in the gap function and even two-gap superconductivity. Our new μSR investigation of LaNiC₂ provides a fresh perspective on the nature of the superconducting pairing state, by yielding the temperature and magnetic field dependencies of the magnetic penetration depth and vortex core size in the mixed state and through a new test of TRSB.