Galaxies form and evolve via a multitude of complex physics. In this work, we investigate the role of cosmic ray (CR) feedback in galaxy evolution and reionisation, by examining its impact on the escape of ionising radiation from galaxies. For this purpose, we present two SPHINX cosmological radiation-magneto-hydrodynamics simulations, enabling, for the first time, a study of the impact of CR feedback on thousands of resolved galaxies during the Epoch of Reionisation (EoR). The simulations differ in their feedback prescriptions: one adopts a calibrated strong supernova (SN) feedback, while the other reduces the strength of SN feedback and includes CR feedback instead. We show that both regulate star formation and match observations of high-redshift UV luminosity functions to a reasonable extent, while also producing a similar amount of hydrogen ionising photons. In contrast to the model with strong SN feedback, the model with CRs lead to incomplete reionisation, which is in strong disagreement with observational estimates of the reionisation history. This is due to CR feedback shaping the ISM differently, filling with gas the low-density cavities carved by SN explosions. As a result, this reduces the escape of ionising photons, at any halo mass, and primarily in the close vicinity of the stars. Our study indicates that CR feedback regulates galaxy growth during the EoR, but negatively affects reionisation. This tension paves the way for the further exploration and refinement of existing galaxy formation and feedback models. Such improvements are crucial in capturing and understanding the process of reionisation and the underlying evolution of galaxies through cosmic time.