The response of a Timepix3 detector with a partial-3D silicon sensor (electrodes partially penetrating the silicon substrate) was studied in a relativistic charged particle beam at the Super-Proton-Synchrotron at the European Center for Particle Physics (CERN). The detector was placed in a mixed ion beam produced by a 330 GeV/c lead ion beam hitting a beryllium target. The spectra of energy loss and 2D tracks from the ions impinging the detector at different angles of incidence are presented and analysed. It was shown that the detector can distinguish ions of charge up to z = 6 for angles of incidence greater than 50°, and ions of charges up to z = 5 when the incident angle was as low as 25°. We observed the volcano effect in the 3D Timepix3 detector, and a maximum of 800 keV per pixel energy was recorded. Charge carriers created in the volume without electrodes were found to have drift times up to 210 ns.