[en] The inclusive K^--meson production in proton-nucleus collisions in the subthreshold energy regime is analyzed in the framework of an appropriate folding model for incoherent primary proton-nucleon and secondary pion-nucleon production processes, which takes properly into account the struck target nucleon momentum and removal energy distribution (nucleon spectral function), novel elementary cross-sections for proton-nucleon reaction channel close to threshold as well as nuclear mean-field potential effects on the one-step and two-step antikaon creation processes. A detailed comparison of the model calculations of the K^- differential cross-sections for the reactions p+⁹Be and p+⁶³Cu at subthreshold energies with the first experimental data obtained at the ITEP proton synchrotron is given, that displays both the relative role of the primary and secondary production channels at considered incident energies and the contributions to the K^- production coming from the use of the single-particle part as well as high-momentum-energy part of the nucleon spectral function. It is found that the pion-nucleon production channel does not dominate in the subthreshold ''hard'' antikaon production in p⁹Be-, p⁶³Cu-collisions and the main contributions to the antikaon yields here come from the direct K^- production mechanism. The influence of the nucleon, kaon and antikaon mean-field potentials on the K^- yield is explored. It is shown that the effect of the nucleon mean-field is of importance in explaining the considered experimental data on ''hard'' antikaon production, whereas the K⁺ and K^- optical potentials play a minor role. The sensitivity of the subthreshold ''soft'' antikaon production in p⁹Be-, p¹²C-reactions to the nucleon, kaon and antikaon mean fields is studied. It is demonstrated that, contrary to the case of ''hard'' antikaon production, the K^- potential has a very strong effect on the K^- yield, which is greater than that from nucleon effective potential. (orig.)