This Letter presents a search for exclusive Z boson production in proton-antiproton collisions at s√=1.96  TeV, using the CDF II detector. No exclusive Z→l+l− candidates are observed and the first upper limit on the exclusive Z cross section in hadron collisions is found to be σexcl(Z)<0.96  pb at 95% confidence level. In addition, eight candidate exclusive dilepton events from the process pp→pγγp→pl+l−p are observed, and a measurement of the cross section for Mll>40  GeV/c2 and |ηl|<4 is found to be σ=0.24+0.13−0.10  pb, which is consistent with the standard model prediction.