[ABRIDGED] We analyse near-infrared integral field spectroscopy of the central starburst region of NGC 1140, obtained at the Gemini-South telescope equipped with CIRPASS. Our ~1.45-1.67 um wavelength coverage includes the bright [Fe II] emission line, as well as high-order Brackett (hydrogen) lines. While strong [Fe II] emission, thought to originate in the thermal shocks associated with supernova remnants, is found throughout the galaxy, both Br 12-4 and Br 14-4 emission, and weak CO(6,3) absorption, is predominantly associated with the northern starburst region. The Brackett lines originate from recombination processes occurring on smaller scales in (young) HII regions. The time-scale associated with strong [Fe II] emission implies that most of the recent star-formation activity in NGC 1140 was induced in the past 35-55 Myr. Based on the spatial distributions of the [Fe II] versus Brackett line emission, we conclude that a galaxy-wide starburst was induced several tens of Myr ago, with more recent starburst activity concentrated around the northern starburst region. This scenario is (provisionally) confirmed by our analysis of the spectral energy distributions of the compact, young massive star clusters (YMCs) detected in new and archival broad-band HST images. The YMC ages in NGC 1140 are all <= 20 Myr, consistent with independently determined estimates of the galaxys starburst age, while there appears to be an age difference between the northern and southern YMC complexes in the sense expected from our CIRPASS analysis. Our photometric mass estimates of the NGC 1140 YMCs, likely upper limits, are comparable to those of the highest-mass Galactic globular clusters and to spectroscopically confirmed masses of (compact) YMCs in other starburst galaxies.
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