Our long-term photometric monitoring of southern nova-like cataclysmic variables with the 1.3-m Small and Moderate Aperture Research Telescope System (SMARTS) telescope found BB Doradus fading from V˜ 14.3 towards a deep low state at V˜ 19.3 in 2008 April. Here we present time-resolved optical spectroscopy of BB Dor in this faint state in 2009. The optical spectrum in quiescence is a composite of a hot white dwarf with Teff= 30 000 ± 5000 K and a M3-M4 secondary star with narrow emission lines (mainly of the Balmer series and He I) superposed. We associate these narrow profiles with an origin on the donor star. An analysis of the radial velocity curve of the Hα emission from the donor star allowed the measurement of an orbital period of 0.154 095 ± 0.000 003 d (3.698 28 ± 0.000 07 h), different from all previous estimates. We detected episodic accretion events which veiled the spectra of both stars and radically changed the line profiles within a time-scale of tens of minutes. This shows that accretion is not completely quenched in the low state. During these accretion episodes the line wings are stronger and their radial velocity curve is delayed by ˜0.2 cycle, similar to that observed in SW Sex and AM Her stars in the high state, with respect to the motion of the white dwarf. Two scenarios are proposed to explain the extra emission: impact of the material on the outer edge of a cold, remnant accretion disc, or the combined action of a moderately magnetic white dwarf (B1≲ 5 MG) and the magnetic activity of the donor star.