Supernovae are stellar explosions driven by gravitational or thermonuclear energy that is observed as electromagnetic radiation emitted over weeks or more. In all known supernovae, this radiation comes from internal energy deposited in the outflowing ejecta by one or more of the following processes: radioactive decay of freshly synthesized elements (typically 56Ni), the explosion shock in the envelope of a supergiant star, and interaction between the debris and slowly moving, hydrogen-rich circumstellar material. Here we report observations of a class of luminous supernovae whose properties cannot be explained by any of these processes. The class includes four new supernovae that we have discovered and two previously unexplained events (SN 2005ap and SCP 06F6) that we can now identify as members of the same class. These supernovae are all about ten times brighter than most type Ia supernova, do not show any trace of hydrogen, emit significant ultraviolet flux for extended periods of time and have late-time decay rates that are inconsistent with radioactivity. Our data require that the observed radiation be emitted by hydrogen-free material distributed over a large radius (~1015 centimetres) and expanding at high speeds (>104 kilometres per second). These long-lived, ultraviolet-luminous events can be observed out to redshifts z>4.