Aims: A detailed study is presented, including estimates of the impact on elemental abundance analysis, of the non-local thermodynamic equilibrium (non-LTE) formation of the high-excitation neutral oxygen 777 nm triplet in model atmospheres representative of stars with spectral types F to K. Methods: We have applied the statistical equilibrium code MULTI to a number of plane-parallel MARCS atmospheric models covering late-type stars (4500 ≤ T_eff ≤ 6500 K, 2 ≤ log g ≤ 5 [cgs], and -3.5 ≤ [Fe/H] ≤ 0). The atomic model employed includes, in particular, recent quantum-mechanical electron collision data. Results: We confirm that the O i triplet lines form under non-LTE conditions in late-type stars, suffering negative abundance corrections with respect to LTE. At solar metallicity, the non-LTE effect, mainly attributed in previous studies to photon losses in the triplet itself, is also driven by an additional significant contribution from line opacity. At low metallicity, the very pronounced departures from LTE are due to overpopulation of the lower level (3s ^5S^o) of the transition. Large line opacity stems from triplet-quintet intersystem electron collisions, a form of coupling previously not considered or seriously underestimated. The non-LTE effects generally become severe for models (both giants and dwarfs) with higher T_eff. Interestingly, in metal-poor turn-off stars, the negative non-LTE abundance corrections tend to rapidly become more severe towards lower metallicity. When neglecting H collisions, they amount to as much as |Δlog ɛ_O| ~ 0.9 dex and ~1.2 dex, respectively at [Fe/H] = -3 and [Fe/H] = -3.5. Even when such collisions are included, the LTE abundance remains a serious overestimate, correspondingly by |Δlog ɛ_O| ~ 0.5 dex and ~0.9 dex at such low metallicities. Although the poorly known inelastic hydrogen collisions thus remain an important uncertainty, the large metallicity-dependent non-LTE effects seem to point to a resulting “low” (compared to LTE) [O/Fe] in metal-poor halo stars. Conclusions: Our results may be important in solving the long-standing [O/Fe] debate. When applying the derived non-LTE corrections, the LTE oxygen abundance inferred from the 777 nm permitted triplet will be decreased substantially at low metallicity. If the classical Drawin formula is employed for O+H collisions, the derived [O/Fe] trend becomes almost flat below [Fe/H] ~ -1, in better agreement with recent literature estimates generally obtained from other oxygen abundance indicators. A value of [O/Fe] ⪉ +0.5 may therefore be appropriate, as suggested by standard theoretical models of type II supernovae nucleosynthetic yields. If neglecting impacts with H atoms instead, [O/Fe] decreases towards lower [Fe/H], which would open new questions. Our tests using ATLAS model atmospheres show that, though non-LTE corrections for metal-poor dwarfs are smaller (by ~0.2 dex when adopting efficient H collisions) than in the MARCS case, our main conclusions are preserved, and that the LTE approach tends to seriously overestimate the O abundance at low metallicity. However, in order to finally reach consistency between oxygen abundances from the different available spectral features, it is of high priority to reduce the large uncertainty regarding H collisions, to undertake a full investigation of the interplay of non-LTE and 3D effects, and to clarify the issue of the temperature scale at low metallicity.