We present Gemini/GNIRS (Gemini Near-Infrared Spectrograph) spectroscopy of the Seyfert 2 galaxy NGC 4388, with simultaneous coverage from 0.85 to 2.5 μm. Several spatially extended emission lines are detected for the first time, both in the obscured and unobscured portion of the optical narrow-line region (NLR), allowing us to assess the combined effects of the central continuum source, outflowing gas and shocks generated by the radio jet on the central 280 pc gas. The H I and [Fe II] lines allow us to map the extinction affecting the NLR. We found that the nuclear region is heavily obscured, with E(B - V) ˜ 1.9 mag. To the NE of the nucleus and up to ˜150 pc, the extinction remains large, ˜1 mag or larger, consistent with the system of dust lanes seen in optical imaging. We derived position-velocity diagrams for the most prominent lines as well as for the stellar component. Only the molecular gas and the stellar component display a well-organized pattern consistent with disc rotation. Other emission lines are kinematically perturbed or show little evidence of rotation. Extended high-ionization emission of sulphur, silicon and calcium is observed to distances of at least 200 pc both NE and SW of the nucleus. We compared flux ratios between these lines with photoionization models and conclude that radiation from the central source alone cannot explain the observed high-ionization spectrum. Shocks between the radio jet and the ambient gas are very likely an additional source of excitation. We conclude that NGC 4388 is a prime laboratory to study the interplay between all these mechanisms.