  A hysteretic in-plane magnetoresistance develops below the superconducting
transition of LaAlO$_3$/SrTiO$_3$ interfaces for $\left|H_{/\!/}\right|<$ 0.15
T, independently of the carrier density or oxygen annealing. We show that this
hysteresis arises from vortex depinning within a thin superconducting layer, in
which the vortices are created by discrete ferromagnetic dipoles located solely
above the layer. We find no evidence for finite-momentum pairing or bulk
magnetism and hence conclude that ferromagnetism is strictly confined to the
interface, where it competes with superconductivity.
