The physics of quantum critical phase transitions connects to some of the most difficult problems in condensed matter physics, including metal–insulator transitions, frustrated magnetism and high-temperature superconductivity. Near a quantum critical point, a new kind of metal emerges, the thermodynamic and transport properties of which do not fit into the unified phenomenology for conventional metals—the Landau Fermi-liquid theory—characterized by a low-temperature limiting T-linear specific heat and a T2 resistivity1. Studying the evolution of the temperature dependence of these observables as a function of a control parameter leads to the identification of both the presence and the nature of the quantum phase transition in candidate systems. In this study we measure the transport properties of BaFe2(As1-xPx)2 below the critical temperature Tc by suppressing superconductivity with high magnetic fields. At sufficiently low temperatures, the resistivity of all compositions (x >0:31) crosses over from a linear to a quadratic temperature dependence, consistent with a low-temperature Fermi-liquid ground state. As compositions with optimal Tc are approached from the overdoped side, this crossover becomes steeper, consistent with models of quantum criticality where the effective Fermi temperature TF goes to zero.