Context. Galaxies, which often contain ionised gas, sometimes also exhibit a so-called low-ionisation nuclear emission line region (LINER). For 30 years, this was attributed to a central mass-accreting supermassive black hole (more commonly known as active galactic nucleus, AGN) of low luminosity, making LINER galaxies the largest AGN sub-population, which dominate in numbers over higher luminosity Seyfert galaxies and quasars. This, however, poses a serious problem. While the inferred energy balance is plausible, many LINERs clearly do not contain any other independent signatures of an AGN. Aims: Using integral field spectroscopic data from the CALIFA survey, we compare the observed radial surface brightness profiles with what is expected from illumination by an AGN. Methods: Essential for this analysis is a proper extraction of emission lines, especially weak lines, such as Balmer Hβ lines, which are superposed on an absorption trough. To accomplish this, we use the GANDALF code, which simultaneously fits the underlying stellar continuum and emission lines. Results: For 48 galaxies with LINER-like emission, we show that the radial emission-line surface brightness profiles are inconsistent with ionisation by a central point-source and hence cannot be due to an AGN alone. Conclusions: The most probable explanation for the excess LINER-like emission is ionisation by evolved stars during the short but very hot and energetic phase known as post-AGB. This leads us to an entirely new interpretation. Post-AGB stars are ubiquitous and their ionising effect should be potentially observable in every galaxy with the gas present and with stars older than ~1 Gyr unless a stronger radiation field from young hot stars or an AGN outshines them. This means that galaxies with LINER-like emission are not a class defined by a property but rather by the absence of a property. It also explains why LINER emission is observed mostly in massive galaxies with old stars and little star formation.