  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 keV 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 neutrino-like excess is found in an exposure of 0.6 t $\times$ y. For the
first time, we use the non-detection of solar neutrinos to constrain the light
yield from 1-2 keV nuclear recoils in liquid xenon, as well as non-standard
neutrino-quark interactions. Finally, we improve upon world-leading constraints
on dark matter-nucleus interactions for dark matter masses between 3 GeV/c$^2$
and 11 GeV/c$^2$ by as much as an order of magnitude.
