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 multiphoton emission probability $[{g}^{2}(0)\ensuremath{\sim}0.10\text{--}0.15]$ at a generation rate up to 10 MHz. We observe Raman photons with linewidths as low as 200 MHz, which is 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.