  Context : If relativistic particle acceleration takes place in colliding-wind
binaries, hard X-rays and gamma-rays are expected through inverse Compton
emission, but to date these have never been unambiguously detected.
  Aims : To detect this emission, observations of Eta Carinae were performed
with INTEGRAL, leveraging its high spatial resolution.
  Methods : Deep hard X-ray images of the region of Eta Car were constructed in
several energy bands.
  Results : The hard X-ray emission previously detected by BeppoSax around Eta
Car originates from at least 3 different point sources. The emission of Eta Car
itself can be isolated for the first time, and its spectrum unambiguously
analyzed. The X-ray emission of Eta Car in the 22-100 keV energy range is very
hard (photon index around 1) and its luminosity is 7E33 erg/s.
  Conclusions : The observed emission is in agreement with the predictions of
inverse Compton models, and corresponds to about 0.1% of the energy available
in the wind collision. Eta Car is expected to be detected in the GeV energy
range.
