We present the spectral energy distribution (SED) between 10 and 200 mu m obtained for the prototype narrow-line X-ray galaxy NGC 7582 with ISOPHOT, the photometer on board the Infrared Space Observatory. The emission is spatially extended and we separated for the first time the nuclear and extranuclear infrared SEDs. The nuclear luminosity (L_IR ~ 4.5*E(10) L_sun) is dominated by cold (T ~ 32 K) dust emission mainly due to star formation activity; warm (T ~ 122 K) dust emission is also present and is probably related to the active nucleus. In addition to a cold component of 30 K, the extranuclear SED (L_IR ~ 1.7*E(10) L_sun) shows emission by colder (T ~ 17 K) dust: this very cold component comprises 90% of the total dust mass of 9.8*E(7) M_sun. Based on observations with ISO, an ESA project with instruments funded by ESA Member States (especially the Principal Investigator countries: France, Germany, Netherlands and the United Kingdom) and with the participation of ISAS and NASA.