  We obtained multi-epoch Very Large Telescope (VLT) optical spectroscopic data
in 2011 and 2012 on the ultraluminous X-ray source (ULX) NGC 5408 X-1. We
confirm that the HeII\lambda4686 line has a broad component with an average
FWHM of v=780\pm64 km/s with a variation of ~13% during observations spanning
over 4 years, and is consistent with the origin in the accretion disc. The
deepest optical spectrum does not reveal any absorption line from a donor star.
Our aim was to measure the radial velocity curve and estimate the parameters of
the binary system. We find an upper limit on the semi-amplitude of the radial
velocity of K=132\pm42 km/s. A search for a periodic signal in the data
resulted in no statistically significant period. The mass function and
constraints on the binary system imply a black hole mass of less than ~510
M_sun. Whilst, a disc irradiation model may imply a black hole mass smaller
than ~431-1985 M_sun, depending on inclination. Our data can also be consistent
with an unexplored orbital period range from a couple of hours to a few days,
thus with a stellar-mass black hole and a subgiant companion.
