  The highly conductive two-dimensional electron gas formed at the interface
between insulating SrTiO$_3$ and LaAlO$_3$ shows low-temperature
superconductivity coexisting with inhomogeneous ferromagnetism. The Rashba
spin-orbit interaction with in-plane Zeeman field of the system favors $p_x \pm
i p_y$-wave superconductivity at finite momentum. Owing to the intrinsic
disorder at the interface, the role of spatial inhomogeneity on the
superconducting and ferromagnetic states becomes important. We find that for
strong disorder, the system breaks up into mutually excluded regions of
superconductivity and ferromagnetism. This inhomogeneity-driven electronic
phase separation accounts for the unusual coexistence of superconductivity and
ferromagnetism observed at the interface.
