ABSTRACT We search for the fastest stars in the subset of stars with radial velocity measurements of the second data release (DR2) of the European Space Agency mission Gaia. Starting from the observed positions, parallaxes, proper motions, and radial velocities, we construct the distance and total velocity distribution of more than 7 million stars in our Milky Way, deriving the full 6D phase space information in Galactocentric coordinates. These information are shared in a catalogue, publicly available at http://home.strw.leidenuniv.nl/~marchetti/research.html. To search for unbound stars, we then focus on stars with a probability greater than $50 $ per cent of being unbound from the Milky Way. This cut results in a clean sample of 125 sources with reliable astrometric parameters and radial velocities. Of these, 20 stars have probabilities greater than 80 per cent of being unbound from the Galaxy. On this latter subsample, we perform orbit integration to characterize the stars’ orbital parameter distributions. As expected given the relatively small sample size of bright stars, we find no hypervelocity star candidates, stars that are moving on orbits consistent with coming from the Galactic Centre. Instead, we find seven hyperrunaway star candidates, coming from the Galactic disc. Surprisingly, the remaining 13 unbound stars cannot be traced back to the Galaxy, including two of the fastest stars (around 700 km s−1). If conformed, these may constitute the tip of the iceberg of a large extragalactic population or the extreme velocity tail of stellar streams.