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 \(\hbox {CaWO}_{4}\) or \(\hbox {Al}_{2}\hbox {O}_{3}\) targets, have discovery potential for dark matter interaction cross sections in the neutrino floor.