Doping is a common route to reducing nanowire transistor on-resistance but has limits. High doping level gives significant loss in gate performance and ultimately complete gate failure. We show that electrolyte gating remains effective even when the Be doping in our GaAs nanowires is so high that traditional metal-oxide gates fail. In this regime we obtain a combination of sub-threshold swing and contact resistance that surpasses the best existing p-type nanowire MOSFETs. Our sub-threshold swing of 75 mV/dec is within 25% of the room-temperature thermal limit and comparable with n-InP and n-GaAs nanowire MOSFETs. Our results open a new path to extending the performance and application of nanowire transistors, and motivate further work on improved solid electrolytes for nanoscale device applications.