We study the spatially resolved excitation properties of the ionized gas in a sample of 646 galaxies using integral field spectroscopy data from the Sloan Digital Sky Survey IV Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) programme. Making use of Baldwin-Philips-Terlevich diagnostic diagrams we demonstrate the ubiquitous presence of extended (kpc scale) low-ionization emission-line regions (LIERs) in both star-forming and quiescent galaxies. In star-forming galaxies LIER emission can be associated with diffuse ionized gas, most evident as extraplanar emission in edge-on systems. In addition, we identify two main classes of galaxies displaying LIER emission: 'central LIER' (cLIER) galaxies, where central LIER emission is spatially extended, but accompanied by star formation at larger galactocentric distances, and 'extended LIER' (eLIER) galaxies, where LIER emission is extended throughout the whole galaxy. In eLIER and cLIER galaxies, LIER emission is associated with radially flat, low Ha equivalent width of line emission (<3 angstrom) and stellar population indices demonstrating the lack of young stellar populations, implying that line emission follows tightly the continuum due to the underlying old stellar population. The H alpha surface brightness radial profiles are always shallower than 1/r(2) and the line ratio [O III] lambda 5007/[O II]lambda lambda 3727,29 (a tracer of the ionization parameter of the gas) shows a flat gradient. This combined evidence strongly supports the scenario in which LIER emission is not due to a central point source but to diffuse stellar sources, the most likely candidates being hot, evolved (post-asymptotic giant branch) stars. Shocks are observed to play a significant role in the ionization of the gas only in rare merging and interacting systems.