Maskin and Riley (Games Econ Behav 45:395–409, 2003) and Lebrun (Games Econ Behav 55:131–151, 2006) prove that the Bayes–Nash equilibrium of first-price auctions is unique. This uniqueness requires the assumption that a buyer never bids above his value (which amounts to the elimination of weakly dominated strategies). We demonstrate that, in asymmetric first-price auctions (with or without a minimum bid), the relaxation of this assumption results in additional equilibria that are substantial. Although in each of these additional equilibria no buyer wins with a bids above his value, the allocation of the object and the selling price may vary among the equilibria. In particular, we show that these yield higher revenue. We show that such phenomena can only occur under certain types of asymmetry in the distributions of values.