  Future dark matter (DM) direct detection searches will be subject to
irreducible neutrino backgrounds that will challenge the identification of an
actual WIMP signal in experiments without directionality sensitivity. We study
the impact of neutrino-quark non-standard interactions (NSI) on this
background, assuming the constraints from neutrino oscillations and the recent
COHERENT experiment data, which are relevant for NSI mediated by light
mediators, $m_{\rm med} \lesssim{\cal O}$(GeV). We calculate the expected
number of neutrino-nucleus elastic scattering events in a Xe-based ton-size
dark matter detector, including solar neutrino fluxes from the $pp$ chain and
CNO cycle as well as sub-GeV atmospheric fluxes and taking into account NSI
effects in both propagation and detection. We find that sizable deviations from
the standard model expectation are possible, but are more pronounced for
flavor-diagonal couplings, in particular for electron neutrinos. We show that
neutrino NSI can enhance or deplete the neutrino-nucleus event rate, which may
impact DM searches in multi-ton detectors.
