Context. The faintest Local Group galaxies found lurking in and around the Milky Way halo provide a unique test bed for theories of structure formation and evolution on small scales. Deep Subaru and Hubble Space Telescope photometry demonstrates that the stellar populations of these galaxies are old and that the star formation activity did not last longer than 2 Gyr in these systems. A few mechanisms that may lead to such a rapid quenching have been investigated by means of hydrodynamic simulations, but these have not provided any final assessment so far. Aims. This is the first in a series of papers aimed at analyzing the roles of stellar feedback, ram pressure stripping, host-satellite tidal interactions, and reionization in cleaning the lowest mass Milky Way companions of their cold gas using high-resolution, three-dimensional hydrodynamic simulations. Methods. We simulated an isolated ultrafaint dwarf galaxy loosely modeled after Boötes I, and examined whether or not stellar feedback alone could drive a substantial fraction of the ambient gas out from the shallow potential well. Results. In contrast to simple analytical estimates, but in agreement with previous hydrodynamical studies, we find that most of the cold gas reservoir is retained. Conversely, a significant amount of the metal-enriched stellar ejecta crosses the boundaries of the computational box with velocities exceeding the local escape velocity and is, thus, likely lost from the system. Conclusions. Although the total energy output from multiple supernova explosions exceeds the binding energy of the gas, no galactic-scale outflow develops in our simulations and as such, most of the ambient medium remains trapped within the weak potential well of the model galaxy. It seems thus unavoidable that to explain the dearth of gas in ultrafaint dwarf galaxies, we will have to resort to environmental effects. This will be the subject of a forthcoming paper.