We report VLA B-configuration observations of the H i 21 cm line on the close disk galaxy pair NGC 5595 and NGC 5597. At the angular resolution of the observations, ∼7.″1 × 4.″2, while most of the H i 21 cm in NGC 5595 and NGC 5597 has the same extent as the optical disk, we have detected for the first time extended structures (streamers) to the northeast (NE) and southwest (SW) of NGC 5595 with no counterparts in blue, red optical (continuum), 20 cm radio continuum, or Hα spectral line emission. One structure is extended by ∼45″ to the NE with blueshifted velocities, and the other is extended by ∼20″ to the SW with redshifted velocities with respect to the systemic velocity. No H i 21 cm emission is detected from the innermost central (nuclear) regions of either galaxy. Lower angular resolution H i 21 cm imaging indicates the nonexistence of any intergalactic hydrogen atomic H i 21 cm gas as tails or bridges between the two galaxies. Our new 20 cm radio continuum emission image of NGC 5597 shows a strong unresolved elongated structure in the central region, in the NE–SW direction, very similar to the spatial location of the innermost Hα spectral line emission. There is no 20 cm continuum emission from its northern spiral arm. In NGC 5595, the 20 cm radio continuum image shows no continuum emission from the NE or SW extended structures with H i 21 cm emission.