  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]
