We present the first signature-based search for delayed photons using an exclusive photon plus missing transverse energy final state. Events are reconstructed in a data sample from the CDF II detector corresponding to 6.3  fb−1 of integrated luminosity from √s=1.96  TeV proton-antiproton collisions. Candidate events are selected if they contain a photon with an arrival time in the detector larger than expected from a promptly produced photon. The mean number of events from standard model sources predicted by the data-driven background model based on the photon timing distribution is 286±24. A total of 322 events are observed. A p value of 12% is obtained, showing consistency of the data with standard model predictions.