  Context: The colliding-wind binary Eta Car exhibits soft X-ray thermal
emission that varies strongly around periastron, and non-thermal emission seen
in hard X-rays and gamma-rays.
  Aims: To definitively identify Eta Car 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.
  Methods: A Chandra observation encompassing the ISGRI error circle was
analysed, and all other soft X-ray sources (including the outer shell of Eta
Car itself) were discarded as likely counter-parts. New hard X-ray images of
Eta Car 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 of 1.8), would produce more emission in the Chandra band than
observed from any point source in the ISGRI error circle apart from Eta Car, as
long as the hydrogen column density to the ISGRI source is lower than 1E24
cm^{-2}. Such sources are rare, thus the ISGRI emission is very likely to be
associated with Eta Car. The eventual contribution of the outer shell to the
non-thermal component also remains fairly limited. Close to periastron, a
3-sigma detection is achieved for the hard X-ray emission of Eta Car, 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 detection can be explained with a hydrogen column
density that does not exceed 6E23 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) appears rather constant as close as a few stellar
radii, well within the acceleration region of the wind.
