The supersymmetric p-branes of type II string theory can be interpreted after compactification as extremal black holes with zero entropy and infinite temperature. We show how the p-branes avoid this apparent, catastrophic instability by developing an infinite mass gap. Equivalently, these black holes behave like elementary particles: they are dressed by effective potentials that prevent absorption of impinging particles. In contrast, configurations with 2, 3, and 4 intersecting branes and their non-extremal extensions, behave increasingly like conventional black holes. These results extend and clarify earlier work by Holzhey and Wilczek in the context of four-dimensional dilaton gravity.