  Stellar streams form through the tidal disruption of satellite galaxies or
globular clusters orbiting a host galaxy. Globular cluster streams are exciting
since they are thin (dynamically cold) and, therefore sensitive to
perturbations from low-mass subhalos. Since the subhalo mass function differs
depending on the dark matter composition, these gaps can provide unique
constraints on dark matter models. However, current samples are limited to the
Milky Way. With its large field of view, deep imaging sensitivity, and high
angular resolution, the upcoming Nancy Grace Roman Space Telescope (Roman)
presents a unique opportunity to increase the number of observed streams and
gaps significantly. This paper presents a first exploration of the prospects
for detecting gaps in streams in M31 and other nearby galaxies with resolved
stars. We simulate the formation of gaps in a Palomar-5-like stream and
generate mock observations of these gaps with background stars in M31 and the
foreground Milky Way stellar fields. We assess Roman's ability to detect gaps
out to 10 Mpc through visual inspection and with the gap-finding tool
${\texttt{FindTheGap}}$. We conclude that gaps of $\approx 1.5$ kpc in streams
that are created from subhalos of masses $\geq5 \times 10^6$M$_{\odot}$ are
detectable within a 2-3 Mpc volume in exposures of 1000s to 1 hour. This volume
contains $\approx 150$ galaxies, including $\approx 8$ galaxies with
luminosities $>10^{9}~$L$_{\odot}$. Large samples of stream gaps in external
galaxies will open up a new era of statistical analyses of gap characteristics
in stellar streams and help constrain dark matter models.
