The investigation of distant galaxy formation and evolution is a powerful tool to constrain dark matter scenarios, supporting and in some cases surpassing other astrophysical and experimental probes. The recent completion of the Hubble Frontier Fields (HFF) programme combining ultra-deep Hubble Space Telescope observations and the magnification power of gravitational lensing produced by foreground galaxy clusters has enabled the detection of the faintest primordial galaxies ever studied. Here we show how the number density of such primordial galaxies allows to constrain a variety of DM models alternative to CDM. In particular, it provides stringent limits on the mass of thermal WDM candidates, on the parameter space of sterile neutrino production models, and on other DM scenarios featuring particles in the keV mass range which is also supported by recent detections of a 3.5keV X-ray line. These constraints are robust and independent of the baryonic physics modeling of galaxy formation and evolution. Fuzzy DM (ultralight DM particles) results strongly disfavored.