Context. η Carinae is the colliding wind binary with the highest mass-loss rate in our Galaxy and the only one in which hard X-ray emission has been detected. Aims. η Carinae is therefore a primary candidate to search for particle acceleration by probing its gamma-ray emission. Methods. We used the first 21 months of Fermi/LAT data to extract gamma-ray (0.2-100 GeV) images, spectra, and light-curves, then combined them with multi-wavelength observations to model the non-thermal spectral energy distribution. Results. A bright gamma-ray source is detected at the position of η Carinae. Its flux at a few 100 MeV can be modelled by an extrapolation of the hard X-ray spectrum towards higher energies. The spectral energy distribution features two distinct components. The first one extends from the keV to GeV energy range, and features an exponential cutoff at ∼1 GeV. It can be understood as inverse Compton scattering of ultraviolet photons by electrons accelerated up to γ ∼ 10 4 in the colliding wind region. The expected synchrotron emission is compatible with the existing upper limit to the non-thermal radio emission. The second component is a hard gamma-ray tail detected above 20 GeV. It could be explained by π 0 -decay of accelerated hadrons interacting with the dense stellar wind. The ratio of the fluxes of the π 0 to inverse Compton components is roughly as predicted by simulations of colliding wind binaries. This hard gamma-ray tail can only be understood if emitted close to the wind collision region. The energy transferred to the accelerated particles (∼5% of the collision mechanical energy) is comparable to that of the thermal X-ray emission. Conclusions. The electron spectrum responsible for the keV to GeV emission was modelled and the observed emission above 20 GeV strongly suggests hadronic acceleration in η Carinae. These observations are thus in good agreement with the colliding wind scenario proposed for η Carinae.