Context. The radio galaxy Cen A has been detected all the way up to the TeV energy range. This raises the question about the dominant emission mechanisms in the high-energy domain. Aims. Spectral analysis allows us to put constraints on the possible emission processes. Here we study the hard X-ray emission, in order to distinguish between a thermal and a non-thermal inverse Compton process. Methods. Using hard X-ray data provided by INTEGRAL, we determined the cut-off of the power-law spectrum in the hard X-ray domain (3−1000 keV). In addition, INTEGRAL data are used to study the spectral variability. The extended emission detected in the gamma-rays by Fermi/LAT is investigated using the data of the spectrometre SPI in the 40−1000 keV range. Results. The hard X-ray spectrum of Cen A shows a significant cut-off at energies E C = 434 +106 −73 keV with an underlying power-law of photon index Γ = 1.73 ± 0.02. A more physical model of thermal Comptonisation (compPS) gives a plasma temperature of kT e = 206 ± 62 keV within the optically thin corona with Compton parameter y = 0.42 +0.09 −0.06 . The reflection component is significant at the 1.9σ level with R = 0.12 +0.09 −0.10 , and a reflection strength R > 0.3 can be excluded on a 3σ level. Time resolved spectral studies show that the flux, absorption, and spectral slope varied in the range f 3−30 keV = 1.2−9.2 × 10 −10 erg cm −2 s −1 , N H = 7−16 × 10 22 cm −2 , and Γ = 1.75−1.87. Extending the cut-off power-law or the Comptonisation model to the gamma-ray range shows that they cannot account for the high-energy emission. On the other hand, a broken or curved power-law model can also represent the data, therefore a non-thermal origin of the X-ray to GeV emission cannot be ruled out. The analysis of the SPI data provides no sign of significant emission from the radio lobes and gives a 3σ upper limit of f 40−1000 keV < ∼ 1.1 × 10 −3 ph cm −2 s −1 . Conclusions. While gamma-rays, as detected by CGRO and Fermi, are caused by non-thermal (jet) processes, the main process in the hard X-ray emission of Cen A is still not unambiguously determined, since it is either dominated by thermal inverse Compton emission or by non-thermal emission from the base of the jet.