Atomically sharp oxide heterostructures exhibit a range of novel physical phenomena that are absent in the parent compounds. A prominent example is the appearance of highly conducting and superconducting states at the interface between LaAlO3 and SrTiO3. Here we report an emergent phenomenon at the LaMnO3/SrTiO3 interface where an antiferromagnetic Mott insulator abruptly transforms into a nanoscale inhomogeneous magnetic state. Upon increasing the thickness of LaMnO3, our scanning nanoSQUID-on-tip microscopy shows spontaneous formation of isolated magnetic nanoislands, which display thermally activated moment reversals in response to an in-plane magnetic field. The observed superparamagnetic state manifests the emergence of thermodynamic electronic phase separation in which metallic ferromagnetic islands nucleate in an insulating antiferromagnetic matrix. We derive a model that captures the sharp onset and the thickness dependence of the magnetization. Our model suggests that a nearby superparamagnetic-ferromagnetic transition can be gate tuned, holding potential for applications in magnetic storage and spintronics.