  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 $50 \%$ 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 $\%$ of being unbound from the Galaxy.
On this latter sub-sample, 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 $7$ hyper-runaway star candidates, coming from the Galactic
disk. Surprisingly, the remaining $13$ unbound stars cannot be traced back to
the Galaxy, including two of the fastest stars (around $700$ km/s). If
conformed, these may constitute the tip of the iceberg of a large extragalactic
population or the extreme velocity tail of stellar streams.
