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Context: The colliding-wind binary Eta Car exhibits soft X-ray thermalemission that varies strongly around periastron, and non-thermal emission seenin hard X-rays and gamma-rays.  Aims: To definitively identify Eta Car as the source of the hard X-rayemission, to examine how changes in the 2-10 keV band influence changes in thehard X-ray band, and to understand more clearly the mechanisms producing thenon-thermal emission using new INTEGRAL observations obtained close toperiastron.  Methods: A Chandra observation encompassing the ISGRI error circle wasanalysed, and all other soft X-ray sources (including the outer shell of EtaCar itself) were discarded as likely counter-parts. New hard X-ray images ofEta Car were studied close to periastron, and compared to previous observationsfar from periastron.  Results: The INTEGRAL component, when represented by a power law (with aphoton index of 1.8), would produce more emission in the Chandra band thanobserved from any point source in the ISGRI error circle apart from Eta Car, aslong as the hydrogen column density to the ISGRI source is lower than 1E24cm^{-2}. Such sources are rare, thus the ISGRI emission is very likely to beassociated with Eta Car. The eventual contribution of the outer shell to thenon-thermal component also remains fairly limited. Close to periastron, a3-sigma detection is achieved for the hard X-ray emission of Eta Car, with aflux similar to the average value far from periastron.  Conclusions: Assuming a single absorption component for both the thermal andnon-thermal sources, this detection can be explained with a hydrogen columndensity that does not exceed 6E23 cm^{-2} without resorting to an intrinsicincrease in the hard X-ray emission. The energy injected in hard X-rays(averaged over a month) appears rather constant as close as a few stellarradii, well within the acceleration region of the wind.

Hard X-ray identification of Eta Carinae and steadiness close to periastron