Polymorphic electronics, in which circuits dynamically alter their function, are a promising approach for secure, fault-tolerant, and adaptive computing systems. A major challenge is developing reconfigurable components that can maintain their new state without requiring constant power. Here, we demonstrate non-volatile doping of graphene field-effect transistors by trapping ions within the electric double layer (EDL) at room temperature using an electric-field-sensitive solid polymer electrolyte. The electrolyte is designed to both facilitate ion motion and undergo the Menshutkin reaction, triggered by the large electric fields generated by the EDL itself. The reaction crosslinks the polyethylene oxide (PEO)-based copolymer and traps ions at the interface between the electrolyte and the channel, giving rise to persistent channel doping that remains after grounding the gate terminal. Applying a positive programming voltage induces stable, n-type doping; however, negative programming voltages induce only weak, counter-doping, underscoring the asymmetry between positive and negative biasing. Low-temperature, dual-gated measurements confirm that ion trapping leads to non-volatility by distinguishing mobile, bulk ions from those trapped at the interface. Programming at ±5 V induces a change in electron sheet carrier density on the order of 1012 cm-2, with only weak dependence on the ion concentration. The programmed states show an average retention of the induced sheet-carrier density of 80% over seven weeks at ambient conditions; the induced charge remains stable over the first week, but starts to fluctuate by ±20% in the last four weeks. This work demonstrates a new solid electrolyte with field-sensitive functionality, and provides a device-level path for the development of electrically reconfigurable logic elements relevant to polymorphic electronics and next-generation computing.