As a part of our ongoing Araucaria Project on the improvement of stellar distance indicators we present deep near-infrared JK imaging of several fields in four Local Group galaxies: LMC, SMC, and the Carina and Fornax dwarf galaxies. These data were obtained under excellent seeing conditions at the European Southern Observatory Very Large Telescope and New Technology Telescope. We determine the mean red clump star magnitudes in the J and K bands in the four galaxies. A comparison of the extinction-corrected K-band red clump star magnitudes with the tip of the red giant branch magnitude, the mean RR Lyrae star V-band magnitude, and the mean K-band magnitude of Cepheid variables at a period of 10 days (for the LMC and SMC) strongly suggests that the red clump star absolute K-band magnitude has a very low (if any) dependence on metallicity over the broad range of metallicities covered by our target galaxies. This finding is in contrast to the mean I- and J-band red clump star magnitudes, which do have a clear metallicity dependence and which we calibrate from our data. Excellent agreement with the former calibration of the red clump I-band magnitude dependence on metallicity of Udalski is found from our new data. We use the Galactic cluster K-band red clump star data of Grocholski & Sarajedini to demonstrate that the K-band red clump star absolute magnitude also has very little (if any) dependence on age over an age range of about 2–8 Gyr. The present study therefore provides clear evidence that the mean K-band magnitude of red clump stars is an excellent distance indicator, with very small (if any) population corrections to be applied over a large range in metallicity and age. Our findings imply that present-day population corrections calculated from models are only accurate at a ±0.15 mag level, which is a great achievement in itself but not accurate enough for high-precision distance scale work. We determine the distances to all our target galaxies from the K-band red clump magnitude, with very small statistical uncertainties. Comparing these distances with those coming from the observed mean I-band magnitudes of the red clump stars, we find evidence that there is likely to be a problem in the photometric calibration of the local, solar neighborhood red clump star K- or I-band magnitudes, which amounts to some 0.2 mag. A redetermination of the absolute photometric calibration of the Hipparcos-observed nearby red clump stars seems necessary to resolve this problem and put the derivation of absolute distances to Local Group galaxies from their red clump stars on a firmer basis.