The identification of sources driving cosmic reionization, a major phase transition from neutral hydrogen to ionized plasma around 600-800 Myr after the Big Bang^1-3, has been a matter of debate^4. Some models suggest that high ionizing emissivity and escape fractions (f_esc) from quasars support their role in driving cosmic reionization^5,6. Others propose that the high f_esc values from bright galaxies generate sufficient ionizing radiation to drive this process^7. Finally, a few studies suggest that the number density of faint galaxies, when combined with a stellar-mass-dependent model of ionizing efficiency and f_esc, can effectively dominate cosmic reionization^8,9. However, so far, comprehensive spectroscopic studies of low-mass galaxies have not been done because of their extreme faintness. Here we report an analysis of eight ultra-faint galaxies (in a very small field) during the epoch of reionization with absolute magnitudes between M_UV ≈ -17 mag and -15 mag (down to 0.005L^⋆ (refs. ^10,11)). We find that faint galaxies during the first thousand million years of the Universe produce ionizing photons with log[ξ_ion (Hz erg^-1)] = 25.80 ± 0.14, a factor of 4 higher than commonly assumed values^12. If this field is representative of the large-scale distribution of faint galaxies, the rate of ionizing photons exceeds that needed for reionization, even for escape fractions of the order of 5%.