Using scanning thermal microscopy we have mapped the spatial distribution of temperatures in an operating nanoscale device formed from a magnetic injector, a Ag connecting wire, and a magnetic detector. An analytical model explained the thermal diffusion over the measured temperature range (2 K -300 K) and injector-detector separation (400 nm -3000 nm). The characteristic diffusion lengths of the Peltier and Joule heat differ remarkably below 60 K, a fact that can be explained by the onset of ballistic phonon heat transfer in the substrate.