Aims. The Gaia mission has provided us full astrometric solutions for over 1.5B sources. However, only the brightest 34M of them have radial velocity measurements. This paper aims to close that gap by obtaining radial velocity estimates from the low-resolution BP/RP spectra that Gaia now provides. These spectra are currently published for about 220M sources, with this number to increase to the full ~2B Gaia sources with Gαία Data Release 4. Methods. To obtain the radial velocity measurements, we fitted Gaia BP/RP spectra with models based on a grid of synthetic spectra. From this we obtained the posterior probability on the radial velocity for each object. Our measured velocities show systematic biases that depend mainly on the colours and magnitudes of stars. We corrected for these effects by using external catalogues of radial velocity measurements. Results. We present a catalogue of about 6.4M sources with our most reliable radial velocity measurements and uncertainties of <300 km s−1 obtained from BP/RP spectra. About 23% of them have no radial velocity measurement from the Gaia RVS. Furthermore, we provide an extended catalogue that contains all 125M sources for which we were able to obtain radial velocity measurements. This catalogue, however, also contains a fraction of measurements for which the reported radial velocities and uncertainties are inaccurate. Conclusions. Although typical uncertainties in the catalogue are significantly higher compared to those obtained with precision spec-troscopy instruments, the number of potential sources to which this method can be applied is orders of magnitude higher than any previous radial velocity catalogue. Further development of the analysis could therefore prove extremely valuable in our understanding of Galactic dynamics.