  Our long-term photometric monitoring of southern nova-like cataclysmic
variables with the 1.3-m SMARTS telescope found BB Doradus fading from V ~ 14.3
towards a deep low state at V ~ 19.3 in April 2008. 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 =
30000 +- 5000 K and a M3-4 secondary star with narrow emission lines (mainly of
the Balmer series and HeI) superposed. We associate these narrow profiles with
an origin on the donor star. Analysis of the radial velocity curve of the
H-alpha emission from the donor star allowed the measurement of an orbital
period of 0.154095 +- 0.000003 d (3.69828 +- 0.00007 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
timescale 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.
