Two-dimensional SrTiO₃-based interfaces stand out among non-centrosymmetric superconductors due to their intricate interplay of gate-tunable Rashba spin-orbit coupling and multi-orbital electronic occupations, whose combination theoretically prefigures various forms of non-standard superconductivity. By employing superconducting transport measurements in nano-devices we present strong experimental indications of unconventional superconductivity in the LaAlO₃/SrTiO₃ interface. The central observations are the substantial anomalous enhancement of the critical current by small magnetic fields applied perpendicularly to the plane of electron motion, and the asymmetric response with respect to the magnetic field direction. These features cannot be accommodated within a scenario of canonical spin-singlet superconductivity. We demonstrate that the experimental observations can be described by a theoretical model based on the coexistence of Josephson channels with intrinsic phase shifts. Our results exclude a time-reversal symmetry breaking scenario and suggest the presence of anomalous pairing components that are compatible with inversion symmetry breaking and multi-orbital physics.