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We report the design, operation and performance of a high-resolution, low-latency, bunch-by-bunch feedback system for nano-beam stabilisation. The system employs novel, ultra-low quality-factor cavity beam position monitors (BPMs), a two-stage analogue signal down-mixing system, and a digital signal processing and feedback board incorporating an FPGA. The FPGA firmware allows for the real-time integration of up to fifteen samples of the BPM waveforms within a measured latency of 232 ns. We show that this real-time sample integration improves significantly the beam position resolution and, consequently, the feedback performance. The best demonstrated real-time beam position resolution was 19 nm, which, as far as we are aware, is the best real-time resolution achieved in any operating BPM system. The feedback was operated in two complementary modes to stabilise the vertical position of the ultra-small beam produced at the focal point of the ATF2 beamline at KEK. In single-BPM feedback mode, beam stabilisation to 50$\pm$5 nm was demonstrated. In two-BPM feedback mode, beam stabilisation to 41$\pm$4 nm was achieved.