The Beijing-Arizona Sky Survey (BASS) is a wide and deep imaging survey to cover a 5400 deg$^2$ area in the Northern Galactic Cap with the 2.3m Bok telescope using two filters ($g$ and $r$ bands). The Mosaic $z$-band Legacy Survey (MzLS) covers the same area in $z$ band with the 4m Mayall telescope. These two surveys will be used for spectroscopic targeting of the Dark Energy Spectroscopic Instrument (DESI). The BASS survey observations were completed in 2019 March. This paper describes the third data release (DR3) of BASS, which contains the photometric data from all BASS and MzLS observations between 2015 January and 2019 March. The median astrometric precision relative to {it Gaia} positions is about 17 mas and the median photometric offset relative to the PanSTARRS1 photometry is within 5 mmag. The median $5sigma$ AB magnitude depths for point sources are 24.2, 23.6, and 23.0 mag for $g$, $r$, and $z$ bands, respectively. The photometric depth within the survey area is highly homogeneous, with the difference between the 20% and 80% depth less than 0.3 mag. The DR3 data, including raw data, calibrated single-epoch images, single-epoch photometric catalogs, stacked images, and co-added photometric catalogs, are publicly accessible at url{http://batc.bao.ac.cn/BASS/doku.php?id=datarelease:home}.
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