A search for exclusive or quasi-exclusive γγ → W+W− production, via pp → p()W+W−p() → p()μ±e∓p() at √s = 8 TeV, is reported using data corresponding to an integrated luminosity of 19.7 fb−1. Events are selected by requiring the presence of an electron-muon pair with large transverse momentum pT(μ±e∓) > 30 GeV, and no associated charged particles detected from the same vertex. The 8 TeV results are combined with the previous 7 TeV results (obtained for 5.05 fb−1 of data). In the signal region, 13 (2) events are observed over an expected background of 3.9 ± 0.6 (0.84 ± 0.15) events for 8 (7) TeV, resulting in a combined excess of 3.4σ over the background-only hypothesis. The observed yields and kinematic distributions are compatible with the standard model prediction for exclusive and quasi-exclusive γγ → W+W− production. Upper limits on the anomalous quartic gauge coupling operators a0,C W (dimension-6) and fM0,1,2,3 (dimension-8), the most stringent to date, are derived from the measured dilepton transverse momentum spectrum.