Compact γ-ray sources are of key importance not only for fundamental research but also for paramount practical applications such as cancer radiotherapy, active interrogation of materials, and high-energy radiography. Particular characteristics are required for meaningful implementation: multi-MeV energies per photon, a high degree of collimation, and a high peak brilliance. Laser-driven sources are theoretically expected to deliver such capabilities but experiments to date have reported either sub-MeV photon energies, or relatively low brilliance. By entering the non-linear regime of Thomson scattering, we report here on the first experimental realisation of a compact laser-driven γ-ray source that simultaneously ensures ultra-high brilliance (≈1019 photons s-1 mm-2 mrad-2 0.1% BW), low divergence (≈ mrad), and high photon energy (up to 18 MeV). The reported brilliance exceeds by two orders of magnitudes those of alternative mechanisms and it is the highest ever achieved in the multi-MeV regime in a laboratory experiment.