Earth, Venus, Mars and asteroids such as Vesta and, perhaps, Lutetia have chondritic bulk compositions with massive silicate mantles surrounding iron cores. Anomalies include Mercury with its abundant metallic iron (about 70% by mass), the Moon with its small iron core, and metal-dominated asteroids. Although a giant impact with proto-Earth can explain the Moon's small core, a giant impact origin for Mercury is problematic. Such a scenario requires that proto-Mercury was blasted apart with far greater specific energy than required for lunar formation, yet retained substantial volatile elements and did not reaccrete its ejected mantle. Here we present numerical hydrocode simulations showing that proto-Mercury could have been stripped of its mantle in one or more high-speed collisions with a larger target planet that survived intact. A projectile that escapes the planet-colliding orbit in this hit-and-run scenario ultimately finds a permanent sink for its stripped mantle silicates. We show that if Mars and Mercury are derived from two planetary embryos that randomly avoided being accreted into a larger body, out of 20 original embryos (the rest having accreted into Venus and Earth), then it is statistically probable that one of those had its mantle stripped in one or two hit-and-run collisions. The same reasoning applies to pairwise accretion of planetesimals in the early Solar System, in which the relic bodies, which avoided becoming accreted, would be expected to have a wide diversity of compositions as a consequence of hit-and-run processes.