Integral field optical spectroscopy with the INTEGRAL system has been used to investigate for the first time the two-dimensional kinematic and ionization properties of the extended, warm, ionized gas in Arp 220 over an area of 75.0"×40.0" (i.e., 28×15 kpc). The structure of the ionized gas is divided into well-identified regions associated with the X-ray-emitting plumes and extended lobes, previously studied in detail by McDowell and collaborators. The overall ionization state of the warm gas in the plumes and lobes, as traced by the [N II]/Hα line, is consistent with high-velocity shocks expanding in a neutral ambient medium. Changes in the ionization state of the gas along the major axis of the plumes are detected, in particular in the outer regions of the northwestern plume, where the transition between the main stellar body of the galaxy and a broad, low surface brightness tidal tail is located. If the plumes are produced by a starburst-driven galactic wind, the efficiency in the conversion of mechanical to radiation energy is a factor of at least 10 smaller than in galactic winds developed in edge-on spiral galaxies with well-defined rotation and axis of outflow. The kinematic properties of the lobes, with an average velocity of +8 km s-1 (east lobe) and -79 km s-1 (west lobe), are to a first order in agreement with the predictions of the merger scenario, according to which the lobes are tidally induced gas condensations produced during the merging process. The largest velocity gradients of 50 km s-1 kpc-1 and velocity deviations of up to +280 and -320 km s-1 from the systemic velocity are associated not with the plumes but with the outer stellar envelope and broad tidal tails at distances of up to 7.5 kpc, indicating that the large-scale kinematics of the extended ionized gas in Arp 220 is most likely dominated by the tidally induced motions, and not by galactic winds associated with nuclear starbursts. Based on observations with the William Herschel Telescope (WHT), operated on the island of La Palma by the Isaac Newton Group (ING) in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.