Finding and characterising the first galaxies that illuminated the early Universe at cosmic dawn is pivotal to understand the physical conditions and the processes that led to the formation of the first stars. In the first few months of operations, imaging from the James Webb Space Telescope (JWST) have been used to identify tens of candidates of galaxies at redshift (z) greater than 10, less than 450 million years after the Big Bang. However, none of these candidates has yet been confirmed spectroscopically, leaving open the possibility that they are actually low-redshift interlopers. Here we present spectroscopic confirmation and analysis of four galaxies unambiguously detected at redshift 10.3<z<13.2, previously selected from NIRCam imaging. The spectra reveal that these primeval galaxies are extremely metal poor, have masses between 10^7 and a few times 10^8 solar masses, and young ages. The damping wings that shape the continuum close to the Lyman edge are consistent with a fully neutral intergalactic medium at this epoch. These findings demonstrate the rapid emergence of the first generations of galaxies at cosmic dawn.