We investigated the dose-response of the external beam therapy 3 (EBT3) films for proton and carbon ion clinical beams, in comparison with conventional radiotherapy beams; we also measured the film response along the energy deposition-curve in water. We performed measurements at three hadrontherapy centres by delivering monoenergetic pencil beams (protons: 63-230 MeV; carbon ions: 115-400 MeV/u), at 0.4-20 Gy dose to water, in the plateau of the depth-dose curve. We also irradiated the films to clinical MV-photon and electron beams. We placed the EBT3 films in water along the whole depth-dose curve for 148.8 MeV protons and 398.9 MeV/u carbon ions, in comparison with measurements provided by a plane-parallel ionization chamber. For protons, the response of EBT3 in the plateau of the depth-dose curve is not different from that of photons, within experimental uncertainties. For carbon ions, we observed an energy dependent under-response of EBT3 film, from 16% to 29% with respect to photon beams. Moreover, we observed an under-response in the Bragg peak region of about 10% for 148.8 MeV protons and of about 42% for 398.9 MeV/u carbon ions. For proton and carbon ion clinical beams, an under-response occurs at the Bragg peak. For carbon ions, we also observed an under-response of the EBT3 in the plateau of the depth-dose curve. This effect is the highest at the lowest initial energy of the clinical beams, a phenomenon related to the corresponding higher LET in the film sensitive layer. This behavior should be properly modeled when using EBT3 films for accurate 3D dosimetry.