  We report constraints on light dark matter (DM) models using ionization
signals in the XENON1T experiment. We mitigate backgrounds with strong event
selections, rather than requiring a scintillation signal, leaving an effective
exposure of $(22 \pm 3)$ tonne-days. Above
$\sim\!0.4\,\mathrm{keV}_\mathrm{ee}$, we observe $<1 \,
\text{event}/(\text{tonne} \times \text{day} \times \text{keV}_\text{ee})$,
which is more than one thousand times lower than in similar searches with other
detectors. Despite observing a higher rate at lower energies, no DM or CEvNS
detection may be claimed because we cannot model all of our backgrounds. We
thus exclude new regions in the parameter spaces for DM-nucleus scattering for
DM masses $m_\chi$ within $3-6\,\mathrm{GeV}/\mathrm{c}^2$, DM-electron
scattering for $m_\chi > 30\,\mathrm{MeV}/\mathrm{c}^2$, and absorption of dark
photons and axion-like particles for $m_\chi$ within $0.186 - 1 \,
\mathrm{keV}/\mathrm{c}^2$.
