We have conducted a deep, complete H i survey, using Arecibo/Arecibo L-band Feed Array (ALFA), of a field centered on the nearby, isolated galaxy, NGC 2903, which is similar to the Milky Way in its properties. The field size was 150 kpc × 260 kpc and the final velocity range spanned from 100 to 1133 km s −1 . The ALFA beams have been mapped as a function of azimuth and cleaned from each azimuth-specific cube prior to forming final cubes. The final H i data are sensitive down to an H i mass of 2 × 10 5 M and column density of 2 × 10 17 cm −2 at the 3 σ 2 δ V level, where σ is the rms noise level and δ V is the velocity resolution. NGC 2903 is found to have an H i envelope that is larger than previously known, extending to at least three times the optical diameter of the galaxy. Our search for companions yields one new discovery with an H i mass of 2.6×10 6 M . The companion is 64 kpc from NGC 2903 in projection, is likely associated with a small optical galaxy of similar total stellar mass, and is dark matter dominated, with a total mass >10 8 M . In the region surveyed, there are now two known companions: our new discovery and a previously known system that is likely a dwarf spheroidal, lacking H i content. If H i constitutes 1% of the total mass in all possible companions, then we should have detected 230 companions, according to Λ cold dark matter (ΛCDM) predictions. Consequently, if this number of dark-matter clumps are indeed present, then they contain less than 1% H i content, possibly existing as very faint dwarf spheroidals or as starless, gasless dark-matter clumps.