We have searched for exclusive gammagamma production in proton-antiproton collisions at sqrt[s]=1.96 TeV, using 532 pb(-1) of integrated luminosity taken by the run II Collider Detector at Fermilab. The event signature requires two electromagnetic showers, each with transverse energy E(T)>5 GeV and pseudorapidity |eta|<1.0, with no other particles detected in the event. Three candidate events are observed. We discuss the consistency of the three events with gammagamma, pi(0)pi(0), or eta eta production. The probability that other processes fluctuate to >or=3 events is 1.7x10(-4). An upper limit on the cross section of pp-->p + gammagamma + p production is set at 410 fb with 95% confidence level.