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High-velocity clouds (HVCs) are interstellar clouds of atomic hydrogen that do not follow normal Galactic rotation and have velocities of a several hundred kilometres per second. A considerable number of these clouds are falling down towards the Galactic disc. HVCs form large and massive complexes, so if they collide with the disc a great amount of energy would be released into the interstellar medium. The cloud-disc interaction produces two shocks: one propagates through the cloud and the other through the disc. The properties of these shocks depend mainly on the cloud velocity and the disc-cloud density ratio. In this work, we study the conditions necessary for these shocks to accelerate particles by diffusive shock acceleration and we study the non-thermal radiation that is produced. We analyse particle acceleration in both the cloud and disc shocks. Solving a time-dependent two-dimensional transport equation for both relativistic electrons and protons, we obtain particle distributions and non-thermal spectral energy distributions. In a shocked cloud, significant synchrotron radio emission is produced along with soft gamma rays. In the case of acceleration in the shocked disc, the non-thermal radiation is stronger; the gamma rays, of leptonic origin, might be detectable with current instruments. A large number of protons are injected into the Galactic interstellar medium, and locally exceed the cosmic ray background. We conclude that under adequate conditions the contribution from HVC-disc collisions to the galactic population of relativistic particles and the associated extended non-thermal radiation might be important.