Context. We report the detection of unidentified infrared (UIR) bands in a filamentary structure associated with Hα emission in the starburst dwarf galaxy NGC 1569 based on imaging and spectroscopic observations of the AKARI satellite. Aims. We investigate the processing and destruction of the UIR band carriers in an outflow from NGC 1569. Methods. We performed observations of NGC 1569 for 6 infrared bands (3.2, 4.1, 7, 11, 15, and 24 μm) with the infrared camera (IRC) onboard AKARI. Near-to mid-infrared (2−13 μm) spectroscopy of a Hα filament was also carried out with the IRC. Results. The extended structure associated with a Hα filament appears bright at 7 μm. Since the IRC 7 μm band (S7) efficiently traces the 6.2 and 7.7 μm UIR band emission, the IRC imaging observations suggest that the filament is bright at the UIR band emission. Follow-up spectroscopic observations with the IRC confirm the presence of 6.2, 7.7, and 11.3 μm emission in the filament. The filament spectrum exhibits strong 11.3 μm UIR band emission relative to the 7.7 μm band compared to the galaxy disk observed with the infrared spectrograph on Spitzer. The near-infrared spectrum also suggests the presence of excess continuum emission in 2.5−5 μm in the filament. Conclusions. The presence of the UIR bands associated with a Hα filament is found by AKARI/IRC observations. The Hα filament is thought to have been formed by the galactic outflow originating from the star-formation activity in the disk of NGC 1569. The destruction timescale of the UIR band carriers in the outflow is estimated to be much shorter (∼1.3 × 10 3 yr) than the timescale of the outflow (∼5.3 Myr). Thus it is unlikely that the band carriers survive the outflow environment. Alternatively, we suggest that the band carriers in the filaments may be produced by the fragmentation of large carbonaceous grains in shocks, which produce the Hα emission. The NIR excess continuum emission cannot be accounted for by free-free emission alone and a hot dust contribution may be needed, although the free-free emission intensity estimated from H i recombination lines has a large uncertainty.