In conventional positron annihilation spectroscopy using radioisotopes, source contributions are unavoidable since positrons annihilate in the material covering the radioisotopes. Part of the positrons annihilate within the radioisotopes even when radioisotopes are deposited directly on a sample. Gamma-ray-induced positron annihilation spectroscopy makes it possible to measure only the spectra of a sample without source contributions since positrons are directly generated inside the sample from the gamma rays by pair production and annihilate inside the same sample. In this study, a new positron age-momentum correlation measurement system using ultrashort pulsed gamma rays is developed. The gamma rays with an energy of 6.6 MeV are generated by the inverse Thomson scattering of laser photons by high-energy electrons and are irradiated to the sample. The laser pulse can fully control the timing of gamma-ray generation. This characteristic and the use of a digital oscilloscope with a 12-bit vertical resolution enable us to develop a simple measurement system. Time-resolved momentum distributions for stainless steel with no defects and deformed interstitial free steel show the explicit differences reflecting the type of defect; for BaF2 single crystals, the results have been interpreted by considering the formation of positronium.