There are several strong K I lines found in the spectra of M dwarfs, among them the doublet near 7700 Å and another doublet near 12 500 Å. We study these optical and near-infrared doublets in a sample of 324 M dwarfs, observed with CARMENES, the high-resolution optical and near-infrared spectrograph at Calar Alto, and investigate how well the lines can be used as photospheric and chromospheric diagnostics. Both doublets have a dominant photospheric component in inactive stars and can be used as tracers of effective temperature and gravity. For variability studies using the optical doublet, we concentrate on the red line component because this is less prone to artefacts from telluric correction in individual spectra. The optical doublet lines are sensitive to activity, especially for M dwarfs later than M5.0 V where the lines develop an emission core. For earlier type M dwarfs, the red component of the optical doublet lines is also correlated with Hα activity. We usually find positive correlation for stars with Hα in emission, while early-type M stars with Hα in absorption show anti-correlation. During flares, the optical doublet lines can exhibit strong fill-in or emission cores for our latest spectral types. On the other hand, the near-infrared doublet lines very rarely show correlation or anti-correlation to Hα and do not change line shape significantly even during the strongest observed flares. Nevertheless, the near-infrared doublet lines show notable resolved Zeeman splitting for about 20 active stars which allows to estimate the magnetic fields B.