A quantum phase transition is an abrupt change between two distinct ground states of a many-body system, driven by an external parameter. In the vicinity of the quantum critical point (QCP) where the transition occurs, a new phase may emerge that is determined by quantum fluctuations and is very different from either phase. In particular, a conducting system may exhibit non-Fermi-liquid behaviour. Although this scenario is well established theoretically, controllable experimental realizations are rare. Here, we experimentally investigate the nature of the QCP in a simple nanoscale system - a spin-polarized resonant level coupled to dissipative contacts. We fine-tune the system to the QCP, realized exactly on-resonance and when the coupling between the level and the two contacts is symmetric. Several anomalous transport scaling laws are demonstrated, including a striking non-Fermi-liquid scattering rate at the QCP, indicating fractionalization of the resonant level into two Majorana quasiparticles. © 2013 Macmillan Publishers Limited.