© 2015 ESO. Context. The Galactic centre is a bright γ-ray source with the GeV-TeV band spectrum composed of two distinct components in the 1-10 GeV and 1-10 TeV energy ranges. The nature of these two components is not clearly understood. Aims. We investigate the γ-ray properties of the Galactic centre to clarify the origin of the observed emission. Methods. We report imaging, spectral, and timing analysis of data from 74 months of observations of the Galactic centre by Fermi/LAT γ-ray telescope complemented by sub-MeV data from approximately ten years of INTEGRAL/PICsIT observations. Results. We find that the Galactic centre is spatially consistent with the point source in the GeV band. The tightest 3σ upper limit on its radius is 0.13° in the 10-300 GeV energy band. The spectrum of the source in the 100 MeV energy range does not have a characteristic turnover that would point to the pion decay origin of the signal. Instead, the source spectrum is consistent with a model of inverse Compton scattering by high-energy electrons. In this a model, the GeV bump in the spectrum originates from an episode of injection of high-energy particles, which happened ~300 years ago. This injection episode coincides with the known activity episode of the Galactic centre region, previously identified using X-ray observations. The hadronic model of source activity could be still compatible with the data if bremsstrahlung emission from high-energy electrons was present in addition to pion decay emission.