In the positron-electron annihilation process, finite deviations from the standard calculation based on Fermi's golden rule are suggested in recent theoretical work. This paper describes an experimental test of the predictions of this theoretical work by searching for events with two photons from positron annihilation of energy larger than the electron rest mass (511 keV). The positrons came from a 22Na source, tagging the third photon from the spontaneous emission of 22Ne∗ de-exitation to suppress backgrounds. Using the collected sample of 1.06 × 107 positron-electron annihilations, triple coincidence photon events in the signal-enhanced energy regions are examined. The observed number of events in two signal regions, NobsSR1 = 0 and NobsSR2 = 0, are, within the current precision, consistent with the expected number of events, NexpSR1 = 0.86 ± 0.08(stat.)+−1.850.81(syst.) and NexpSR2 = 0.37 ± 0.05(stat.)+−0.800.29(syst.) from Fermi's golden rule, respectively. Based on the P(d) modeling, a 90% CL lower limit on the photon wave packet size is obtained.