We present several lines of evidence, based on different kinds of observations, and we conclude that it is likely that rotational fission has occurred for a fraction of the known trans-Neptunian objects (TNOs). It is also likely that a number of binary systems have formed from that process in the trans-Neptunian belt. We show that Haumea is, potentially, an example of an object that has suffered rotational fission. Its current fast spin would be a slight evolution of a primordial fast spin, rather than the result of a catastrophic collision. This is because the percentage of objects rotating faster than 4 h would not be small in a Maxwellian distribution of spin rates, which fits the current TNO rotation data base. Besides, the specific total angular momentum of Haumea and its satellites falls close to that of the high-size-ratio asteroid binaries, which are thought to be the result of rotational fission or mass shedding. We also present N-body simulations of rotational fission applied to the case of Haumea. These show that this process is feasible; it might have generated satellites, and it might have even created a 'family' of bodies orbitally associated to Haumea. The orbitally associated bodies might come from the direct ejection of fragments, according to our simulations, or through the evolution of a proto-satellite formed during the fission event. The disruption of an escaped fragment after the fission might also create the orbitally related bodies. If any of these mechanisms are correct, other rotational fission families could be detectable in the trans-Neptunian belt in the future. Perhaps, TNO pairs might even be found (i.e. pairs of bodies sharing very similar orbital elements but not bound together).