  The early third data release (EDR3) of the European Space Agency satellite
Gaia provides coordinates, parallaxes, and proper motions for ~1.47 billion
sources in our Milky Way, based on 34 months of observations. The combination
of Gaia DR2 radial velocities with the more precise and accurate astrometry
provided by Gaia EDR3 makes the best dataset available to search for the
fastest nearby stars in our Galaxy. We compute the velocity distribution of ~7
million stars with precise parallaxes, to investigate the high-velocity tail of
the velocity distribution of stars in the Milky Way. We release a catalogue
with distances, total velocities, and corresponding uncertainties for all the
stars considered in our analysis, available at
https://sites.google.com/view/tmarchetti/research . By applying quality cuts on
the Gaia astrometry and radial velocities, we identify a clean subset of 94
stars with a probability Pub > 50% to be unbound from our Galaxy. 17 of these
have Pub > 80% and are our best candidates. We propagate these stars in the
Galactic potential to characterize their orbits. We find that 11 stars are
consistent with being ejected from the Galactic disk, and are possible
hyper-runaway star candidates. The other 6 stars are not consistent with coming
from a known star-forming region. We investigate the effect of adopting a
parallax zero point correction, which strongly impacts our results: when
applying this correction, we identify only 12 stars with Pub > 50%, 3 of these
having Pub > 80%. Spectroscopic follow-ups with ground-based telescopes are
needed to confirm the candidates identified in this work.
