  Context. We investigate the stellar population and the origin of diffuse
light around brightest cluster galaxies.
  Aims. We study the stellar population of the dynamically hot stellar halo of
NGC 3311, the brightest galaxy in the Hydra I cluster, and that of photometric
substructures in the diffuse light to constrain the origin of these components.
  Methods. We analyze absorption lines in medium-resolution, long-slit spectra
in the wavelength range 4800-5800 angstrom obtained with FORS2 at the Very
Large Telescope. We measure the equivalent width of Lick indices out to 20 kpc
from the center of NGC 3311 and fit them with stellar population models that
account for the [alpha/Fe] overabundance.
  Results. Stars in the dynamically hot halo of NGC 3311 are old (age > 13
Gyr), metal-poor ([Z/H] ~ -0.35), and alpha-enhanced ([alpha/Fe] ~ 0.48).
Together with the high velocity dispersion, these measurements indicate that
the stars in the halo were accreted from the outskirts of other early-type
galaxies, with a possible contribution from dwarf galaxies. We identify a
region in the halo of NGC 3311 associated with a photometric substructure where
the stellar population is even more metal-poor ([Z/H] ~ -0.73). In this region,
our measurements are consistent with a composite stellar population superposed
along the line of sight, consisting of stars from the dynamically hot halo of
NGC 3311 and stars stripped from dwarf galaxies. The latter component
contributes < 28% to the local surface brightness.
  Conclusions. The build-up of diffuse light around NGC 3311 is on-going. Based
on the observed stellar population properties, the dominant part of these stars
may have come from the outskirts of bright early-type galaxies, while stars
from stripped dwarf galaxies are presently being added.
