Until now, organic semiconductors have failed to achieve high performance in highly integrated, sub-100 nm transistors. Consequently, single-crystalline materials such as single-walled carbon nanotubes, MoS2 or inorganic semiconductors are the materials of choice at the nanoscale. Here we show, using a vertical field-effect transistor design with a channel length of only 40 nm and a footprint of 2 × 80 × 80 nm2, that high electrical performance with organic polymers can be realized when using electrolyte gating. Our organic transistors combine high on-state current densities of above 3 MA cm-2, on/off current modulation ratios of up to 108 and large transconductances of up to 5,000 S m-1. Given the high on-state currents at such large on/off ratios, our novel structures also show promise for use in artificial neural networks, where they could operate as memristive devices with sub-100 fJ energy usage.