High-resolution 21 cm H I deep fields provide spatially and kinematically resolved images of neutral hydrogen at different redshifts, which are key to understanding galaxy evolution across cosmic time and testing predictions of cosmological simulations. Here we present results from a pilot for an H I deep field done with the Karl G. Jansky Very Large Array (VLA). We take advantage of the newly expanded capabilities of the telescope to probe the redshift interval 0 <z <0.193 in one observation. We observe the COSMOS field for 50 hr, which contains 413 galaxies with optical spectroscopic redshifts in the imaged field of 34' x 34' and the observed redshift interval. We have detected neutral hydrogen gas in 33 galaxies in different environments spanning the probed redshift range, including three without a previously known spectroscopic redshift. The detections have a range of H I and stellar masses, indicating the diversity of galaxies we are probing. We discuss the observations, data reduction, results, and highlight interesting detections. We find that the VLA's B-array is the ideal configuration for H I deep fields since its long spacings mitigate radio frequency interference. This pilot shows that the VLA is ready to carry out such a survey, and serves as a test for future H I deep fields planned with other Square Kilometer Array pathfinders.