We have imaged the ultraluminous infrared galaxy Arp 220 in light of the near-infrared [Fe II] 1.257 mu m and Pa beta lines, and have obtained spectra in the J- and H-band atmospheric windows. Arp 220 is a strong source of [Fe II] and Pa beta emission, with luminosities of 1.3 x 10(41) and 9.2 x 10(40) ergs s(-1), respectively. The [Fe II] and Pa beta emission are both extended over the central 2''-3'', but with different morphologies. The Pa beta line is strongly peaked at the position of the western nucleus seen at 2.2 mu m (Graham et al. 1990) with a fainter ''spur'' in the direction of the eastern nucleus. The [Fe II] emission line shows a weak peak at the western nucleus along with diffuse emission extending to the east, but with no indication of a secondary maximum. The [Fe II] is more extended in the north-south direction than the Pa beta line. Nearly 75% of the detected [Fe II] emission is spatially resolved. The overall [Fe II]-to-Pa beta line flux ratio in Arp 220 is consistent with that seen over similar spatial scales in Seyfert 2 galaxies, yet larger than what is measured in galaxies with nuclear starbursts. The [Fe II]-to-Pa beta line flux ratio is spatially variable being approximately 0.8 at the position of the western nucleus and approximately 2.0 at radii of up to 500 pc. We suggest that the extended [Fe II] emission is produced through the interaction of fast shocks with ambient gas in the ISM at the base of the outflowing, supernovae-driven superwind mapped by Heckman et al. (1987). The bolometric luminosity of the starburst required to power this wind is estimated to be at least 2 x 10(11) L.. If the spatially unresolved [Fe II] emission is produced via a large number of supernova remnants, the implied rate is similar to 0.6 yr(-1). The overall luminosity of such a starburst could account for a large fraction (1/2-1/3) of the Arp 220 energy budget, but the large deficit of ionizing photons (as counted by the Pa beta luminosity) requires that the starburst be rapidly declining and/or have a low upper mass cutoff. Alternatively, dust may effectively compete with the gas for ionizing photons, or much of the ionizing radiation may escape through ''holes'' in the ISM. It is also possible that a buried AGN produces a large fraction of the unresolved [Fe II] and Pa beta emission. We briefly discuss these possibilities in light of these new imaging and spectroscopic data.