Aims. We evaluate non-local thermodynamical equilibrium (non-LTE) line formation for the two ions of iron and check the ionization equilibrium between Fe i and Fe ii in model atmospheres of the cool reference stars based on the best available complete model atom for neutral and singly-ionized iron.Methods. We present a comprehensive model atom for Fe with more than 3000 measured and predicted energy levels. As a test and first application of the improved model atom, iron abundances are determined for the Sun and five stars with well determined stellar parameters and high-quality observed spectra. The efficiency of inelastic collisions with hydrogen atoms in the statistical equilibrium of iron is empirically estimated from inspection of their different influence on the Fe i and Fe ii lines in the selected stars. Results. Non-LTE leads to systematically depleted total absorption in the Fe i lines and to positive abundance corrections in agreement with the previous studies, however, the magnitude of such corrections is smaller compared to the earlier results. These non-LTE corrections do not exceed 0.1 dex for the solar metallicity and mildly metal-deficient stars, and they vary within 0.21 dex and 0.35 dex in the very metal-poor stars HD 84937 and HD 122563, respectively, depending on the assumed efficiency of collisions with hydrogen atoms. Based on the analysis of the Fe i/Fe ii ionization equilibrium in these two stars, we recommend to apply the Drawin formalism in non-LTE studies of Fe with a scaling factor of 0.1. For the Fe ii lines non-LTE corrections do not exceed 0.01 dex in absolute value over the whole range of stellar parameters that are considered. This study reveals two problems. The first one is that g f -values available for the Fe i and Fe ii lines are not accurate enough to pursue high-accuracy absolute stellar abundance determinations. For the Sun, the mean non-LTE abundance obtained from 54 Fe i lines is 7.56 ± 0.09 and the mean abundance from 18 Fe ii lines varies between 7.41 ± 0.11 and 7.56 ± 0.05 depending on the source of the g f -values. The second problem is that lines of Fe i give, on average, a 0.1 dex lower abundance compared with those of Fe ii lines for HD 61421 and HD 102870, even when applying a differential line-by-line analysis with regard to the Sun. A disparity between neutral atoms and first ions points to problems of stellar atmosphere modelling or/and effective temperature determination.