In this paper we consider the production of proton-antiproton pairs in two-photon interactions in electron-positron and heavy-ion collisions. We try to understand the dependence of the total cross section on the photon-photon c.m. energy as well as corresponding angular distributions measured by the Belle Collaboration for the γγ → pp process. To understand the Belle data we include the proton-exchange, the f 2 (1270) and f 2 (1950) s-channel exchanges, as well as the handbag mechanism. The helicity amplitudes for the γγ → f 2 → pp process are written explicitly based on a Lagrangian approach. The parameters of vertex form factors are adjusted to the Belle data. Having described the angular distributions for the γγ → pp process we present first predictions for the ultraperipheral, ultrarelativistic, heavy-ion reaction 208 Pb 208 Pb → 208 Pb 208 Pb pp. Both, the total cross section and several differential distributions for experimental cuts corresponding to the ALICE, ATLAS, CMS, and LHCb experiments are presented. We find the total cross section 100 µb for the ALICE cuts, 160 µb for the ATLAS cuts, 500 µb for the CMS cuts, and 104 µb taking into account the LHCb cuts. This opens a possibility to study the γγ → pp process at the LHC. 25.75.Dw,13.60.Rj,13.90.+i