  A commonly encountered obstacle in indirect searches for galactic dark matter
is how to disentangle possible signals from astrophysical backgrounds. Given
that such signals are most likely subdominant, the search for pronounced
spectral features plays a key role for indirect detection experiments;
monochromatic gamma-ray lines or similar features related to internal
bremsstrahlung, in particular, provide smoking gun signatures. We perform a
dedicated search for the latter in the data taken by the Fermi gamma-ray space
telescope during its first 43 months. To this end, we use a new adaptive
procedure to select optimal target regions that takes into account both
standard and contracted dark matter profiles. The behaviour of our statistical
method is tested by a subsampling analysis of the full sky data and found to
reproduce the theoretical expectations very well. The limits on the dark matter
annihilation cross-section that we derive are stronger than what can be
obtained from the observation of dwarf galaxies and, at least for the model
considered here, collider searches. While these limits are still not quite
strong enough to probe annihilation rates expected for thermally produced dark
matter, future prospects to do so are very good. In fact, we already find a
weak indication, with a significance of 3.1\sigma (4.3\sigma) when (not) taking
into account the look-elsewhere effect, for an internal bremsstrahlung-like
signal that would correspond to a dark matter mass of \sim150 GeV; the same
signal is also well fitted by a gamma-ray line at around 130 GeV. Although this
would be a fascinating possibility, we caution that a much more dedicated
analysis and additional data will be necessary to rule out or confirm this
option.
