The existence of dark matter is probably one of the fundamental mysteries of modern science and unraveling its nature has become one of the primary goals of modern Physics. Despite representing 85% of all matter in the Universe, we do not know what it is. In its simplest description, it is made up of particles that interact with each other and with ordinary matter only through gravity. However, this description does not correspond to any physical model. Finding out what dark matter is requires finding evidence of some kind of interaction of dark matter that goes beyond gravity. In our work, we show that we have discovered what could be the first clear evidence in this regard. The distribution of dark matter in sufficiently small galaxies only depends on the nature of the dark matter. An original technique has allowed us to confirm that the distribution of dark matter in galaxies with only a thousand solar masses requires self-interacting dark matter. The degree of interaction is so small that it does not substantially modify the current cosmological model, but the existence of this self-interaction should play a fundamental role as a diagnostic tool to identify which physical models of dark matter are viable or not.