GRB 160821B is a short duration gamma-ray burst (GRB) detected and localized by the Neil Gehrels Swift Observatory in the outskirts of a spiral galaxy at z = 0.1613, at a projected physical offset of 16 kpc from the galaxy's center. We present X-ray, optical/nIR, and radio observations of its counterpart and model them with two distinct components of emission: a standard afterglow, arising from the interaction of the relativistic jet with the surrounding medium, and a kilonova, powered by the radioactive decay of the sub-relativistic ejecta. Broadband modelling of the afterglow data reveals a weak reverse shock propagating backward into the jet, and a likely jet-break at 3.5 d. This is consistent with a structured jet seen slightly off-axis (theta(view) similar to theta(core)) while expanding into a low-density medium (n approximate to 10(-3) cm(-3)). Analysis of the kilonova properties suggests a rapid evolution towards red colours, similar to AT2017gfo, and a low-nIR luminosity, possibly due to the presence of a long-lived neutron star. The global properties of the environment, the inferred low mass (M-ej less than or similar to 0.006 M-circle dot) and velocities (v(ej) greater than or similar to 0.05c) of lanthanide-rich ejecta are consistent with a binary neutron star merger progenitor.