We report results from searches for new physics with low-energy electronic recoil data recorded with the XENON1T detector. With an exposure of 0.65 tonne-years and an unprecedentedly low background rate of $76\pm 2_{\mathrm{stat}}\mathrm{events}/(\mathrm{tonne}\times \mathrm{year}\times \mathrm{keV})$ between 1 and 30 keV, the data enable one of the most sensitive searches for solar axions, an enhanced neutrino magnetic moment using solar neutrinos, and bosonic dark matter. An excess over known backgrounds is observed at low energies and most prominent between 2 and 3 keV. The solar axion model has a $3.4\sigma$ significance, and a three-dimensional 90% confidence surface is reported for axion couplings to electrons, photons, and nucleons. This surface is inscribed in the cuboid defined by $g_{\mathrm{ae}}<3.8\times {10}^{-12}$, $g_{\mathrm{ae}}{g}_{\mathrm{an}}^{\mathrm{eff}}<4.8\times {10}^{-18}$, and $g_{\mathrm{ae}}{g}_{a\gamma }<7.7\times {10}^{-22}{\mathrm{GeV}}^{-1}$, and excludes either $g_{\mathrm{ae}}=0$ or $g_{\mathrm{ae}}{g}_{a\gamma }=g_{\mathrm{ae}}{g}_{\mathrm{an}}^{\mathrm{eff}}=0$. The neutrino magnetic moment signal is similarly favored over background at $3.2\sigma$, and a confidence interval of ${\mu }_{\nu }\in (1.4,2.9)\times {10}^{-11}{\mu }_B$ (90% C.L.) is reported. Both results are in strong tension with stellar constraints. The excess can also be explained by $\beta$ decays of tritium at $3.2\sigma$ significance with a corresponding tritium concentration in xenon of $(6.2\pm 2.0)\times {10}^{-25}\mathrm{mol}/\mathrm{mol}$. Such a trace amount can neither be confirmed nor excluded with current knowledge of its production and reduction mechanisms. The significances of the solar axion and neutrino magnetic moment hypotheses are decreased to $2.0\sigma$ and $0.9\sigma$, respectively, if an unconstrained tritium component is included in the fitting. With respect to bosonic dark matter, the excess favors a monoenergetic peak at $(2.3\pm 0.2)\mathrm{keV}$ (68% C.L.) with a $3.0\sigma$ global ($4.0\sigma$ local) significance over background. This analysis sets the most restrictive direct constraints to date on pseudoscalar and vector bosonic dark matter for most masses between 1 and $210\mathrm{keV}/c^2$. We also consider the possibility that ${}^{37}\mathrm{Ar}$ may be present in the detector, yielding a 2.82 keV peak from electron capture. Contrary to tritium, the ${}^{37}\mathrm{Ar}$ concentration can be tightly constrained and is found to be negligible.