Direct dark matter detection experiments based on a liquid xenon target are leading the search for dark matter particles with masses above ∼5 GeV/c2, but have limited sensitivity to lighter masses because of the small momentum transfer in dark matter-nucleus elastic scattering. However, there is an irreducible contribution from inelastic processes accompanying the elastic scattering, which leads to the excitation and ionization of the recoiling atom (the Migdal effect) or the emission of a bremsstrahlung photon. In this Letter, we report on a probe of low-mass dark matter with masses down to about 85 MeV/c2 by looking for electronic recoils induced by the Migdal effect and bremsstrahlung using data from the XENON1T experiment. Besides the approach of detecting both scintillation and ionization signals, we exploit an approach that uses ionization signals only, which allows for a lower detection threshold. This analysis significantly enhances the sensitivity of XENON1T to light dark matter previously beyond its reach.