Thermal photons radiated in heavy-ion collisions represent an important signal for a recently discovered new state of matter, the deconfined quark-gluon plasma. However, a clean identification of this signal requires precise knowledge of the prompt photons produced simultaneously in hard collisions of quarks and gluons, mostly through their fragmentation. In this paper, we demonstrate that PHENIX data on photons produced in proton-proton collisions with low transverse momenta allow one to extract new information on this fragmentation process. While existing data do not yet convincingly favor one parameterization (BFG II) over the two other frequently used photon fragmentation functions (BFG I and GRV NLO), the data sets recorded by PHENIX and STAR at BNL RHIC in 2013 with tenfold higher statistics should allow for such an analysis.