It is theoretically expected that a supermassive black hole (SMBH) in the centre of a typical nearby galaxy disrupts a solar-type star every ∼105 yr, resulting in a bright flare lasting for months. Sgr A*, the resident SMBH of the Milky Way, produces (by comparison) tiny flares that last only hours but occur daily. Here we explore the possibility that these flares could be produced by disruption of smaller bodies – asteroids. We show that asteroids passing within an au of Sgr A* could be split into smaller fragments which then vaporize by bodily friction with the tenuous quiescent gas accretion flow on to Sgr A*. The ensuing shocks and plasma instabilities may create a transient population of very hot electrons invoked in several currently popular models for Sgr A* flares, thus producing the required spectra. We estimate that asteroids larger than ∼10 km in size are needed to power the observed flares, with the maximum possible luminosity of the order of 1039 erg s−1. Assuming that the asteroid population per parent star in the central parsec of the Milky Way is not too dissimilar from that around stars in the solar neighbourhood, we estimate the asteroid disruption rates, and the distribution of the expected luminosities, finding a reasonable agreement with the observations. We also note that planets may be tidally disrupted by Sgr A* as well, also very infrequently. We speculate that one such disruption may explain the putative increase in Sgr A* luminosity ∼100 yr ago.