Context. The colliding-wind binary η Carinae exhibits soft X-ray thermal emission that varies strongly around the periastron passage. It has been found to have non-thermal emission, thanks to its detection in hard X-rays using INTEGRAL and Suzaku, and also in γ-rays with AGILE and Fermi. Aims. This paper attempts to definitively identify η Carinae as the source of the hard X-ray emission, to examine how changes in the 2-10 keV band influence changes in the hard X-ray band, and to understand more clearly the mechanisms producing the non-thermal emission using new INTEGRAL observations obtained close to periastron passage. Methods. To strengthen the identification of η Carinae with the hard X-ray source, a long Chandra observation encompassing the INTEGRAL/ISGRI error circle was analysed, and all other soft X-ray sources (including the outer shell of η Carinae itself) were discarded as likely counter-parts. To expand the knowledge of the physical processes governing the X-ray lightcurve, new hard X-ray images of η Carinae were studied close to periastron, and compared to previous observations far from periastron. Results. The INTEGRAL component, when represented by a power law (with a photon index Γ of 1.8), would produce more emission in the Chandra band than observed from any point source in the ISGRI error circle apart from η Carinae, as long as the hydrogen column density to the ISGRI source is lower than N H 10 24 cm −2 . Sources with such a high absorption are very rare, thus the hard X-ray emission is very likely to be associated with η Carinae. The eventual contribution of the outer shell to the non-thermal component also remains fairly limited. Close to periastron passage, a 3-σ detection is achieved for the hard X-ray emission of η Carinae, with a flux similar to the average value far from periastron. Conclusions. Assuming a single absorption component for both the thermal and non-thermal sources, this 3-σ detection can be explained with a hydrogen column density that does not exceed N H 6 × 10 23 cm −2 without resorting to an intrinsic increase in the hard X-ray emission. The energy injected in hard X-rays (averaged over a month timescale) appears to be rather constant at least as close as a few stellar radii, well within the acceleration region of the wind.