Conducting polymers are mixed ionic–electronic conductors that are emerging candidates for neuromorphic computing, bioelectronics and thermoelectrics. However, fundamental aspects of their many-body correlated electron–ion transport physics remain poorly understood. Here we show that in p-type organic electrochemical transistors it is possible to remove all of the electrons from the valence band and even access deeper bands without degradation. By adding a second, field-effect gate electrode, additional electrons or holes can be injected at set doping states. Under conditions where the counterions are unable to equilibrate in response to field-induced changes in the electronic carrier density, we observe surprising, non-equilibrium transport signatures that provide unique insights into the interaction-driven formation of a frozen, soft Coulomb gap in the density of states. Our work identifies new strategies for substantially enhancing the transport properties of conducting polymers by exploiting non-equilibrium states in the coupled system of electronic charges and counterions.