  We report the discovery of 11 very faint (r< 23), low surface brightness
({\mu}_r< 27 mag/arcsec^2) dwarf galaxies in one deep field in the Virgo
cluster, obtained by the prime focus cameras (LBC) at the Large Binocular
Telescope (LBT). These extend our previous sample to reach a total number of 27
galaxies in a field of just of 0.17 deg^2 located at a median distance of 390
kpc from the cluster center. Their association with the Virgo cluster is
supported by their separate position in the central surface brightness - total
magnitude plane with respect to the background galaxies of similar total
magnitude. For a significant fraction (26\%) of the sample the association to
the cluster is confirmed by spectroscopic follow-up. We show that the mere
abundance of satellite galaxies corresponding to our observed number in the
target field provides extremely tight constraints on Dark Matter models with
suppressed power spectrum compared to the Cold Dark Matter case, independently
of the galaxy luminosity distribution. In particular, requiring the observed
number of satellite galaxies not to exceed the predicted abundance of Dark
Matter sub-halos yields a limit m_X >3 keV at 1-{\sigma} and m_X > 2.3 keV at
2-{\sigma} confidence level for the mass of thermal Warm Dark Matter particles.
Such a limit is competitive with other limits set by the abundance of
ultra-faint satellite galaxies in the Milky Way, is completely independent of
baryon physics involved in galaxy formation, and has the potentiality for
appreciable improvements with next observations. We extend our analysis to Dark
Matter models based on sterile neutrinos, showing that our observations set
tight constraints on the combination of sterile neutrino mass m_{\nu} and
mixing parameter sin^2(2{\theta}). We discuss the robustness of our results
with respect to systematics.
