The International Technology Roadmap for Semiconductors challenges the device research community to reduce the transistor footprint containing all components to 40 nanometers within the next decade. We report on a p-channel transistor scaled to such an extremely small dimension. Built on one semiconducting carbon nanotube, it occupies less than half the space of leading silicon technologies, while delivering a significantly higher pitch-normalized current density - above 0.9 milliampere per micrometer at a low supply voltage of 0.5 volts with a subthreshold swing of 85 millivolts per decade. Furthermore, we show transistors with the same small footprint built on actual high-density arrays of such nanotubes that deliver higher current than that of the best-competing silicon devices under the same overdrive, without any normalization. We achieve this using low-resistance end-bonded contacts, a high-purity semiconducting carbon nanotube source, and self-assembly to pack nanotubes into full surface-coverage aligned arrays.