The cores of some small planetesimals, such as asteroid (16) Psyche, are thought to have been exposed through collisions during the early solar system that removed their mantles. These small bodies likely solidified from the top down representing a fundamentally different solidification regime to that of Earth's core. Here we derive simplified models of the downwards solidification of the metallic crust, and consider thermal convection and the potential for viscous delamination of the weak, warm base of the crust to provide a buoyancy flux sufficient to drive a dynamo. Thermal buoyancy is very short lived (~1000 years), and therefore cannot be the source of measured paleomagnetic remanence. In contrast, viscous delamination is found to provide a long-lasting buoyancy flux sufficient to generate an intense, multipolar magnetic field, while not greatly affecting the crustal solidification time. Our results suggest that a Psyche-sized (150 km radius) body solidified in roughly 6.7 - 20 Myr, and that delamination produced a strong magnetic field over much of this time. Finally, including light, insoluble impurities, such as sulfur, results in a partially solid mushy zone at the base of the crust. This further weakens the base of the crust and results in smaller scale delamination events. Despite a significant change in the dynamics of delamination, the time to total solidification and the predicted properties of the magnetic field are broadly comparable to the sulfur-free case, though we argue this may result in observable compositional stratification of the body.