We have measured the equivalent width WK of the K i resonance line at 7699 Angstroms for a large sample of low activity late-type stars observed with high spectral resolution and we have verified that the relation WK vs. Teff is monotonically decreasing, for both dwarf and giant stars. This behaviour is different from that of the Na I D lines for stars of the same type, which showed that the relation WNa vs. Teff has a maximum for Teff ~ 4000 K, which is better defined for giants than for dwarfs (Tripicchio et al. 1997). The fit of the observed K I equivalent widths by means of a NLTE spectral line synthesis using conventional background opacity shows that, for dwarf stars, the adopted models overestimate the observed WK for temperatures <~ 4000 K. This result is similar to that discussed for the Na I D lines in our previous paper. On the other hand, for giant stars with Teff <~ 3800 K these models in general underestimate WK. The discrepancies between observed and computed WK and WNa for cool stars are much stronger than the variations due to uncertainties in either atmospheric model or line parameters, like effective temperature and surface gravity, or Van der Waals broadening. For M dwarf stars, the most convincing explanation for the disagreement is the lack of atomic and molecular line opacity in the adopted models. In fact, a NLTE spectral synthesis including an additional background opacity reproduces with a good level of accuracy the equivalent widths, as well as the general shape of the profiles for both the Na I D and K I lines, in a subsample of early-M dwarfs. Based on observations collected at the European Southern Observatory (ESO), La Silla, Chile, and at the McDonald Observatory, Mt. Locke, Texas, USA