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 \hbox{$E_{\rm C} = 434 {+106 \atop -73}$}EC=434-73+106 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 kTe =  206 ± 62 keV within the optically thin corona with Compton parameter \hbox{$y = 0.42 {+0.09 \atop -0.06}$}y=0.42-0.06+0.09. The reflection component is significant at the 1.9σ level with \hbox{$R = 0.12 {+0.09 \atop -0.10}$}R=0.12-0.10+0.09, 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 f3−30 keV = 1.2−9.2 × 10-10   erg   cm-2 s-1, NH = 7−16 × 1022   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 f40−1000 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.