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Atmospheric and solar neutrinos scattering coherently off target nuclei will be an important background source for the next generation of direct dark matter searches. In this work we focus on calcium tungstate as target material. For comparison with existing works we calculate the neutrino floor indicating which sensitivities can be reached before the neutrino background appears. In addition, we investigate the sensitivities of future direct dark matter searches using CRESST-II like detectors. Extending previous works we take into account achievable energy resolutions and thresholds as well as beta and gamma backgrounds for this investigation. We show that an exploration of WIMP-nucleon cross sections below the neutrino floor is possible for exposures of ≳10 kg-years. In the third part we show that a first detection of coherent neutrino nucleus scattering of atmospheric and solar neutrinos using the same detectors and the backgrounds is feasible for exposures of ≳50 kg-years.