We present spectroscopy from the Keck telescope of three sets of objects in the Hubble Deep Field (HDF) that have lens-like morphologies. In the case of J123641+621204, which is composed of four objects with similar colors and a mean separation of less than or similar to 0.'' 8, we find at least two distinct components at redshifts of z = 3.209 and z = 3.220 that are separated by 0.'' 5 spatially. Each of these components has narrow Ly alpha emission and possibly N V emission and Si IV and C IV in absorption or with a P Cygni profile. The second case is J123652+621227, which has an arclike feature offset by 1.'' 8 to the southwest of a red elliptical-like galaxy and a ''counterimage'' offset 1.'' 4 on the opposite side. We tentatively find a single line at 5301 Angstrom at the spatial position of the counterimage and no corresponding emission line at the position of the are. The colors of the counterimage are consistent with the identification of this line as Ly alpha at z = 3.36. The colors of the are are different than those of the counterimage, and thus both the colors and spectra indicate that this object is unlikely to be a gravitational lens. For a third lensing candidate (J123656+621221), which is a blue are offset by 0.'' 9 from a red, elliptical-like galaxy, our spectroscopy does not clearly resolve the system spatially, which complicates the interpretation of the spectrum. We discuss possible identifications of a number of absorption features and a very tentative detection of a pair of emission lines at 5650 Angstrom and 5664 Angstrom, and we find that gravitational lensing remains a possibility in this case. We conclude that the frequency of strong gravitational lensing by galaxies in the HDF appears to be very low. This result is difficult to reconcile with the introduction of a cosmological constant to account for the large number of faint blue galaxies via a large volume element at high redshift and tends to favor models in which very faint galaxies are at fairly modest redshifts.