The production of the ηc(1S) state in proton-proton collisions is probed via its decay to the p anti-p final state with the LHCb detector, in the rapidity range 2.0 < y < 4.5 and in the meson transverse-momentum range p_T > 6.5 GeV/c. The cross-section for prompt production of ηc(1S) mesons relative to the prompt J/ψ cross-section is measured, for the first time, to be σ_ηc(1S)/σ_J/ψ = 1.74 ± 0.29 ± 0.28 ± 0.18_B at a centre-of-mass energy √s = 7 TeV using data corresponding to an integrated luminosity of 0.7 fb^−1, and σ_ηc(1S)/σ_J/ψ = 1.60 ± 0.29 ± 0.25 ± 0.17_B at √s = 8 TeV using 2.0 fb^−1. The uncertainties quoted are, in order, statistical, systematic, and that on the ratio of branching fractions of the ηc(1S) and J/ψ decays to the p anti-p final state. In addition, the inclusive branching fraction of b-hadron decays into ηc(1S) mesons is measured, for the first time, to be B(b → ηc X) = (4.88 ± 0.64 ± 0.29 ± 0.67_B) × 10^−3, where the third uncertainty includes also the uncertainty on the J/ψ inclusive branching fraction from b-hadron decays. The difference between the J/ψ and ηc(1S) meson masses is determined to be 114.7 ± 1.5 ± 0.1 MeV/c^2.