28 pages, 23 figures, 3 tablesLepton colliders are considered as options to complement and to extend the physics programme at the Large Hadron Collider. The Compact Linear Collider (CLIC) is an $e^+e^-$ collider under development aiming at centre-of-mass energies of up to 3 TeV. For experiments at CLIC, a hadron sampling calorimeter with tungsten absorber is proposed. Such a calorimeter provides sufficient depth to contain high-energy showers, while allowing a compact size for the surrounding solenoid. A fine-grained calorimeter prototype with tungsten absorber plates and scintillator tiles read out by silicon photomultipliers was built and exposed to particle beams at CERN. Results obtained with electrons, pions and protons of momenta up t...