We investigate the experimental data, total and differential cross sections, on the production of K + -mesons in pA interactions at projectile energies between Tp = 0.8 and 2.9 GeV, covering the transition across the free nucleon-nucleon threshold at 1.58 GeV. No clear evidence for the expected change of the dominant reaction mechanism from two-step to direct kaon production is found. It is suggested that further data, in particular at forward angles, are taken in order to clarify the situation. It also is shown that, independent of the beam energy and emission angle, the invariant K + -production cross sections show an overall exponential scaling behaviour with the squared four-momentum transfer between the beam proton and the produced K + -meson for t<−0.05 GeV 2 . The most recent data from COSY-Jülich, differential cross sections measured for t>0 GeV 2 , show a strongly different t dependence. Further data at forward angles and different beam energies are needed in order to exploit this region of kinematically extreme conditions.