We have searched for prompt production of χc1, χc2 and X(3872) in continuum e+e- annihilations using a 386fb-1 data sample collected around s=10.6GeV with the BABAR detector using the γJ/ψ decay mode. After accounting for the feed-down from ψ(2S)→γχc1,2, no significant signal for prompt χc1,2 production is observed. We present improved upper limits at 90% confidence level on the production cross sections of 77fb for χc1 and 79fb for χc2, for events where the χc momentum exceeds 2.0 GeV and there are at least three additional charged tracks. These limits are consistent with NRQCD predictions. We also set an upper limit on the prompt production of X(3872) through the decay X(3872)→γJ/ψ. © 2007 The American Physical Society.