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 $sim16rm{arcmin}^{2}$ and $sim180rm{arcmin}^{2}$ and reach median rms values of $0.41murm{Jy,beam}^{-1}$ and $0.53murm{Jy,beam}^{-1}$, 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 $5sigma$. 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 $sim2.8mu$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.
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