  We present results from the search for astrometric accelerations of stars in
$\omega$ Centauri using 13 years of regularly-scheduled {\it Hubble Space
Telescope} WFC3/UVIS calibration observations in the cluster core. The
high-precision astrometry of $\sim$160\,000 sources was searched for
significant deviations from linear proper motion. This led to the discovery of
four cluster members and one foreground field star with compelling acceleration
patterns. We interpret them as the result of the gravitational pull by an
invisible companion and determined preliminary Keplerian orbit parameters,
including the companion's mass. {For the cluster members} our analysis suggests
periods ranging from 8.8 to 19+ years and dark companions in the mass range of
$\sim$0.7 to $\sim$1.4$M_\mathrm{sun}$. At least one companion could exceed the
upper mass-boundary of white dwarfs and can be classified as a neutron-star
candidate.
