  Coherent elastic neutrino-nucleus scattering (CE$\nu$NS) poses an irreducible
background in the search for dark matter-nucleus elastic scatterings, which is
commonly known as the neutrino floor. As direct dark matter search experiments
keep improving their sensitivity into so far unexplored regions, they face the
challenge of approaching this neutrino floor. A precise description of the
CE$\nu$NS signal is therefore crucial for the description of backgrounds for
future DM searches. In this work we discuss the scenario of detecting neutrinos
in low-threshold, high-exposure cryogenic solid state experiments optimized for
the search of low-mass dark matter. The energy range considered is completely
dominated by solar neutrinos. In absence of any dark matter events, we treat
solar neutrinos as the main signal of interest. We show that sensitivity to the
flux of neutrinos from different production mechanisms can be achieved. In
particular we investigate the sensitivity to the flux of pp and $^{7}$Be
neutrinos, as well as CNO neutrinos. Furthermore, we investigate the
sensitivity to dark matter signals in the presence of a solar neutrino
background for different experimental scenarios, which are defined by three
parameters: the target material, the energy threshold and the exposure. We show
that experiments with thresholds of $\mathcal{O}$(eV) and exposures of
$\mathcal{O}$(tonne-years), using CaWO$_{4}$ or Al$_{2}$O$_{3}$ targets, have
discovery potential for dark matter interaction cross sections in the neutrino
floor.
