A transient in the Local Group dwarf irregular galaxy NGC 6822 (Barnard's Galaxy) was discovered on 2017 August 2 and is only the second classical nova discovered in that galaxy. We conducted optical, near-ultraviolet, and X-ray follow-up observations of the eruption, the results of which we present here. This 'very fast' nova had a peak $V$-band magnitude in the range $-7.41>M_V>-8.33$ mag, with decline times of $t_{2,V} = 8.1 \pm 0.2$ d and $t_{3,V} = 15.2 \pm 0.3$ d. The early- and late-time spectra are consistent with an Fe II spectral class. The H$\alpha$ emission line initially has a full width at half-maximum intensity of $\sim 2400$ km s$^{-1}$ - a moderately fast ejecta velocity for the class. The H$\alpha$ line then narrows monotonically to $\sim1800$ km s$^{-1}$ by 70 d post-eruption. The lack of a pre-eruption coincident source in archival Hubble Space Telescope imaging implies that the donor is a main sequence, or possibly subgiant, star. The relatively low peak luminosity and rapid decline hint that AT 2017fvz may be a 'faint and fast' nova.