The realisation of a GaN high voltage vertical p–n diode operating at >3.9 kV breakdown with a specific on-resistance <0.9 mΩ cm² is reported. Diodes achieved a forward current of 1 A for on-wafer, DC measurements, corresponding to a current density >1.4 kA/cm². An effective critical electric field of 3.9 MV/cm was estimated for the devices from analysis of the forward and reverse current–voltage characteristics. This suggests that the fundamental limit to the GaN critical electric field is significantly greater than previously believed.