We report our identification of the optical afterglow and host galaxy of the short-duration gamma-ray burst GRB 160821B. The spectroscopic redshift of the host is $z=0.162$, making it one of the lowest redshift sGRBs identified by Swift. Our intensive follow-up campaign using a range of ground-based facilities as well as HST, XMM and Swift, shows evidence for a late-time excess of optical and near-infrared emission in addition to a complex afterglow. The afterglow light-curve at X-ray frequencies reveals a narrow jet, $theta_jsim1.9^{+0.10}_{-0.03}$ deg, that is refreshed at $>1$ day post-burst by a slower outflow with significantly more energy than the initial outflow that produced the main GRB. Observations of the 5 GHz radio afterglow shows a reverse shock into a mildly magnetised shell. The optical and near-infrared excess is fainter than AT2017gfo associated with GW170817, and is well explained by a kilonova with dynamic ejecta mass $M_{rm dyn}=(1.0pm0.6)times10^{-3}$ M$_{odot}$ and a secular (postmerger) ejecta mass with $M_{rm pm}=(1.0pm0.6)times10^{-2}$ M$_odot$, consistent with a binary neutron star merger resulting in a short-lived massive neutron star. This optical and near-infrared dataset provides the best-sampled kilonova light-curve without a gravitational wave trigger to date.
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