As differentiated planetesimals cool, their cores can solidify from the outside in1, as evidenced by palaeomagnetic measurements and cooling-rate estimates of iron meteorites2,3. The details of outside-in solidification and fate of residual core melt are poorly understood. For a core primarily composed of iron and nickel alloyed with lighter constituent elements such as sulfur, this inward core growth would probably be achieved by growth of solid iron-nickel dendrites4. Growth of iron-nickel dendrites results in interconnected pockets of residual melt that become progressively enriched in sulfur up to a eutectic composition of 31 wt% sulfur as iron-nickel continues to solidify4. Here, we show that regions of residual sulfur-enriched iron-nickel melt in the core attain sufficient excess pressures to propagate via dykes into the mantle. Thus, core material will intrude into the overlying rocky mantle or possibly even erupt onto the planetesimal's surface. We refer to these processes collectively as ferrovolcanism. Our calculations show that ferrovolcanic surface eruptions are more likely on bodies with mantles less than 50 km thick. We show that intrusive ferromagmatism can produce pallasites, an enigmatic class of meteorites composed of olivine crystals entrained in a matrix of iron-nickel metal4. Ferrovolcanic eruptions may explain the observations that asteroid 16 Psyche has a bulk density inconsistent with iron meteorites5 yet shows evidence of a metallic surface composition6.