[en] We analyzed the K⁺ production at projectile energies below the free nucleon-nucleon threshold in proton-nucleus collisions and studied the contribution from different reaction channels to this process. It was shown that the secondary pion-nucleon (π + N → Λ + K⁺) channel significantly prevails over the direct (p + N → Λ + N + K⁺) one. The momentum and angular K⁺ spectra were calculated in the framework of different production channels in order to propose experimental measurements which allow to distinguish the mechanisms discussed. We also studied K⁺ d correlations and considered the deuteron reabsorption in the nuclear medium as well as the fast deuteron background. The features of K⁺ p correlations related to the different production channels were analyzed as well. (orig.)