We report on a search for nuclear recoil signals from solar ^{8}B neutrinos elastically scattering off xenon nuclei in XENON1T data, lowering the energy threshold from 2.6 to 1.6 keV. We develop a variety of novel techniques to limit the resulting increase in backgrounds near the threshold. No significant ^{8}B neutrinolike excess is found in an exposure of 0.6 t×y. For the first time, we use the nondetection of solar neutrinos to constrain the light yield from 1-2 keV nuclear recoils in liquid xenon, as well as nonstandard neutrino-quark interactions. Finally, we improve upon world-leading constraints on dark matter-nucleus interactions for dark matter masses between 3 and 11 GeV c^{-2} by as much as an order of magnitude.