  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.
