  We report on a fast, bandwidth-tunable single-photon source based on an
epitaxial GaAs quantum dot. Exploiting spontaneous spin-flip Raman transitions,
single photons at $780\,$nm are generated on-demand with tailored temporal
profiles of durations exceeding the intrinsic quantum dot lifetime by up to
three orders of magnitude. Second-order correlation measurements show a low
multi-photon emission probability ($g^{2}(0)\sim\,0.10-0.15$) at a generation
rate up to $10\,$MHz. We observe Raman photons with linewidths as low as
$200\,$MHz, narrow compared to the $1.1\,$GHz linewidth measured in resonance
fluorescence. The generation of such narrow-band single photons with controlled
temporal shapes at the rubidium wavelength is a crucial step towards the
development of an optimized hybrid semiconductor-atom interface.
