The relative biological effectiveness (RBE) of protons under conditions suitable for eye treatment has been studied. A complete three-dimensional modelling of the beam delivery system has been performed. Proton Monte Carlo transport calculations have been performed to obtain the proton energy distributions at different positions in a water phantom including the influence of range shifter, modulator wheel, scattering foils and collimators. A beam with a kinetic energy of 68 MeV +/- 250 keV has been simulated with respect to the HMI-Berlin eye treatment facility. The dependence of the RBE on absorbed dose and position within a spread-out Bragg peak (SOBP) has been investigated with the track structure model. Due to a decreasing proton energy with depth, the energy transfer per pathlength within the SOBP increases, affecting the RBE. An RBE increasing with depth as well as with decreasing absorbed dose has been found for the endpoint inactivation of V79 and CH2B2 hamster cells. RBE values at the distal end of the SOBP up to 1.3 and 1.5 have been found at a dose of 14 Gy and 2 Gy respectively. Within the SOBP plateau, no lateral variation of RBE has been found for a given depth. The model used offers the possibility of introducing a variable RBE in treatment planning.