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 \mu m , [ FeII ] 1.64 \mu m and Br \gamma narrow-band filters . We obtained high-quality spatial maps of HeI 1.08 \mu m , [ FeII ] 1.64 \mu m and Br \gamma emission across the galaxy , and used them together with HST /ACS images of NGC 1569 in the H \alpha filter to derive the two-dimensional spatial map of the dust extinction and surface star formation rate density . We show that dust extinction ( as derived from the H \alpha /Br \gamma flux ratio ) is rather patchy and , on average , higher in the North-West ( NW ) portion of the galaxy [ E _ { g } ( B-V ) \simeq 0.71 mag ] than in the South-East [ E _ { g } ( B-V ) \simeq 0.57 mag ] . Similarly , the surface density of star formation rate ( computed from either the dereddened H \alpha or dereddened Br \gamma image ) peaks in the NW region of NGC 1569 , reaching a value of about 4 \times 10 ^ { -6 } M _ { \odot } yr ^ { -1 } pc ^ { -2 } . The total star formation rate as estimated from the integrated , dereddened H \alpha ( or , alternatively , Br \gamma ) luminosity is about 0.4 M _ { \odot } yr ^ { -1 } , and the total supernova rate from the integrated , dereddened [ FeII ] 1.64 \mu m luminosity is about 0.005 yr ^ { -1 } ( assuming a distance of 3.36 Mpc ) . The azimuthally averaged [ FeII ] 1.64 \mu m/Br \gamma 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 H \alpha and HeI 1.08 \mu m luminosities suggests that the latest burst of star formation occurred about 4 Myr ago and produced new stars with a total mass of \simeq 1.8 \times 10 ^ { 6 } M _ { \odot } .