  We compute the mass function of galactic dark matter halos for different
values of the Warm Dark Matter (WDM) particle mass m_X and compare it with the
abundance of ultra-faint galaxies derived from the deepest UV luminosity
function available so far at redshift z~2. The magnitude limit M_UV=-13 reached
by such observations allows us to probe the WDM mass functions down to scales
close to or smaller than the half-mass mode mass scale ~10^9 M_sun. This
allowed for an efficient discrimination among predictions for different m_X
which turn out to be independent of the star formation efficiency adopted to
associate the observed UV luminosities of galaxies to the corresponding dark
matter masses. Adopting a conservative approach to take into account the
existing theoretical uncertainties in the galaxy halo mass function, we derive
a robust limit m_X>1.8 keV for the mass of thermal relic WDM particles when
comparing with the measured abundance of the faintest galaxies, while m_X>1.5
keV is obtained when we compare with the Schechter fit to the observed
luminosity function. The corresponding lower limit for sterile neutrinos
depends on the modeling of the production mechanism; for instance m_sterile > 4
keV holds for the Shi-Fuller mechanism. We discuss the impact of observational
uncertainties on the above bound on m_X. As a baseline for comparison with
forthcoming observations from the HST Frontier Field, we provide predictions
for the abundance of faint galaxies with M_UV=-13 for different values of m_X
and of the star formation efficiency, valid up to z~4.
