We study the contribution of thermal and non-thermal processes to the inverse Compton emission of the radio galaxy M87 by modelling its broad-band emission. Through this we aim to derive insight into where within the AGN the X-ray, γ-ray and VHE emission is produced. We have analysed all available INTEGRAL IBIS/ISGRI (Imager on Board INTEGRAL Spacecraft/INTEGRAL Soft Gamma-Ray Imager) data on M87, spanning almost 10 years, to set an upper limit to the average hard X-ray flux of f(20-60 keV) ≲ 3 × 10−12  erg cm−2 s−1, using several techniques beyond the standard analysis which are also presented here. We also analysed hard X-ray data from Suzaku/PIN taken late 2006 November, and we report the first hard X-ray detection of M87 with a flux of f(20-60 keV) = 10−11 erg cm−2 s−1. In addition we analyse data from Fermi/Large Area Telescope, INTEGRAL/Joint European Monitor in X-rays, and Suzaku/X-ray Imaging Spectrometer. We collected historical radio/IR/optical and VHE data and combined them with the X-ray and γ-ray data, to create broad-band spectral energy distributions (SEDs) for the average low-flux state and the flaring state. The resulting SEDs are modelled by applying a single-zone SSC model with a jet angle of θ = 15°. We also show that modelling the core emission of M87 using a single-zone synchrotron self-Compton model does represent the SED, suggesting that the core emission is dominated by a BL Lac-type AGN core. Using SED modelling we also show that the hard X-ray emission detected in 2006 is likely due to a flare of the jet knot HST-1, rather than being related to the core.