We experimentally investigate the mechanism of formation of self-assembled arrays of nanoislands surrounding dopant sources on the (001) surface of yttria-stabilized zirconia. Initially, we used lithographically defined thin-film patches of gadolinia-doped ceria (GDC) as dopant sources. During annealing at approximately one-half the melting temperature of zirconia, surface diffusion of dopants leads to the breakup of the surface around the source, creating arrays of epitaxial nanoislands with a characteristic size (~100 nm) and alignment along elastically compliant directions, <110>. The breakup relieves elastic strain energy at the expense of increasing surface energy. On the basis of understanding the mechanism of island formation, we introduce a simple and versatile powder-based doping process for spontaneous surface patterning. The new process bypasses lithography and conventional vapor-source doping, opening the door to spontaneous surface patterning of functional ceramics and other refractory materials. In addition to using GDC solid-solution powders, we demonstrate the effectiveness of the process in another system based on Eu2O3.