Galaxies in the local Universe are known to follow bimodal distributions in the global stellar populations properties. We analyze the distribution of the local average stellar-population ages of 654,053 sub-galactic regions resolved on ~1-kpc scales in a volume-corrected sample of 394 galaxies, drawn from the CALIFA-DR3 integral-field-spectroscopy survey and complemented by SDSS imaging. We find a bimodal local-age distribution, with an old and a young peak primarily due to regions in early-type galaxies and star-forming regions of spirals, respectively. Within spiral galaxies, the older ages of bulges and inter-arm regions relative to spiral arms support an internal age bimodality. Although regions of higher stellar-mass surface-density, mu*, are typically older, mu* alone does not determine the stellar population age and a bimodal distribution is found at any fixed mu*. We identify an "old ridge" of regions of age ~9 Gyr, independent of mu*, and a "young sequence" of regions with age increasing with mu* from 1-1.5 Gyr to 4-5 Gyr. We interpret the former as regions containing only old stars, and the latter as regions where the relative contamination of old stellar populations by young stars decreases as mu* increases. The reason why this bimodal age distribution is not inconsistent with the unimodal shape of the cosmic-averaged star-formation history is that i) the dominating contribution by young stars biases the age low with respect to the average epoch of star formation, and ii) the use of a single average age per region is unable to represent the full time-extent of the star-formation history of "young-sequence" regions. ; Comment: 17 pages, 11 figures, MNRAS accepted