  We present ultra-deep, matched-resolution Karl G. Jansky Very Large Array
(VLA) observations at 10 and $3$ GHz in the COSMOS field: the COSMOS-XS survey.
The final 10 and $3$ GHz images cover $\sim16\rm{arcmin}^{2}$ and
$\sim180\rm{arcmin}^{2}$ and reach median rms values of
$0.41\mu\rm{Jy\,beam}^{-1}$ and $0.53\mu\rm{Jy\,beam}^{-1}$, respectively. Both
images have an angular resolution of $\sim 2.0''$. To fully account for the
spectral shape and resolution variations across the broad bands, we image all
data with a multi-scale, multi-frequency synthesis algorithm. We present source
catalogs for the 10 and $3$ GHz image with 91 and 1498 sources, respectively,
above a peak brightness threshold of $5\sigma$. We present source counts with
completeness corrections included that are computed via Monte Carlo
simulations. Our corrected radio counts at $3$ GHz with direct detections down
to $\sim2.8\mu$Jy are consistent within the uncertainties with other results at
3 and 1.4 GHz, but extend to fainter flux densities than previous direct
detections. The ultra-faint $3$ GHz number counts are found to exceed the
counts predicted by the semi-empirical radio sky simulations developed in the
framework of the SKA Simulated Skies project, consistent with previous P(D)
analyses. Our measured source counts suggest a steeper luminosity function
evolution for these faint star-forming sources. The semi-empirical Tiered Radio
Extragalactic Continuum Simulation (T-RECS) predicts this steeper evolution and
is in better agreement with our results. The $10$ GHz radio number counts also
agree with the counts predicted by the T-RECS simulation within the expected
variations from cosmic variance. In summary, the multi-band, matched-resolution
COSMOS-XS survey in the well-studied COSMOS field provides a high-resolution
view of the ultra-faint radio sky that can help guide next generation radio
facilities.
