Proton and carbon ion beams present advantageous depth-dose distributions with respect to X-rays. Carbon ions allow a better control of "radioresistant" tumours due to their higher biological response. For deep-seated tumours proton and carbon ion beams of some nA and energies of about 200 MeV and 400 MeV/u respectively are needed. For these applications TERA proposed the "cyclinac": a high-frequency linac which boosts the hadrons accelerated by a cyclotron. The dimensions of the complex can be reduced if higher accelerating gradients are achieved in the linac. To test the maximum achievable fields, a 3 GHz cavity has been built by TERA. The 19 mm-long cell is foreseen to be excited at 200 Hz by 3 us RF pulses and should reach a 40 MV/m accelerating gradient, which corresponds to a peak surface electric field Es of 260 MV/m. In a first high-power test performed at CTF3 the cell was operated at 50 Hz with a maximum peak power of 1 MW. The maximum Es achieved was above 350 MV/m. The breakdown rate at these field values was around 10-1 bpp/m. The maximum value of the modified Poynting vector is close to the best values achieved by high gradient structures at 12 and 30 GHz.