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We used the near-IR imager/spectrograph LUCIFER mounted on the Large Binocular Telescope (LBT) to image, with sub-arcsec seeing, the local dwarf starburst NGC 1569 in the JHK bands and HeI 1.08 micron, [FeII] 1.64 micron and Brgamma narrow-band filters. We obtained high-quality spatial maps of HeI, [FeII] and Brgamma emission across the galaxy, and used them together with HST/ACS images of NGC 1569 in the Halpha filter to derive the two-dimensional spatial map of the dust extinction and surface star formation rate density. We show that dust extinction is rather patchy and, on average, higher in the North-West (NW) portion of the galaxy [E_g(B-V) = 0.71 mag] than in the South-East [E_g(B-V) = 0.57 mag]. Similarly, the surface density of star formation rate peaks in the NW region of NGC 1569, reaching a value of about 4 x 10^-6 M_sun yr^-1 pc^-2. The total star formation rate as estimated from the integrated, dereddened Halpha luminosity is about 0.4 M_sun yr^-1, and the total supernova rate from the integrated, dereddened [FeII] luminosity is about 0.005 yr^-1 (assuming a distance of 3.36 Mpc). The azimuthally averaged [FeII]/Brgamma flux ratio is larger at the edges of the central, gas-deficient cavities (encompassing the super star clusters A and B) and in the galaxy outskirts. If we interpret this line ratio as the ratio between the average past star formation (as traced by supernovae) and on-going activity (represented by OB stars able to ionize the interstellar medium), it would then indicate that star formation has been quenched within the central cavities and lately triggered in a ring around them. The number of ionizing hydrogen and helium photons as computed from the integrated, dereddened Halpha and HeI luminosities suggests that the latest burst of star formation occurred about 4 Myr ago and produced new stars with a total mass of ~1.8 x 10^6 M_sun. [Abridged]