We present the stellar population analysis of a sample of 12 dwarf elliptical galaxies, observed with the SAURON integral field unit, using the full-spectrum fitting method. We show that star formation histories (SFHs) resolved into two populations can be recovered even within a limited wavelength range, provided that high signal-to-noise ratio (S/N) data are used. We confirm that dEs have had complex SFHs, with star formation extending to (more) recent epochs: for the majority of our galaxies star formation activity was either still strong a few (≲5) Gyr ago or they experienced a secondary burst of star formation roughly at that time. This latter possibility is in agreement with the proposed dE formation scenario where tidal harassment drives the gas remaining in their progenitors inwards and induces a star formation episode. For one of our field galaxies, ID 0918, we find a correlation between its stellar population and kinematic properties, pointing to a possible merger origin of its kinematically decoupled core. One of our cluster objects, VCC 1431, appears to be composed exclusively of an old population (≳10-12 Gyr). Combining this with our earlier dynamical results, we conclude that the galaxy was either ram-pressure stripped early on in its evolution in a group environment and subsequently tidally heated, or that it evolved in situ in the cluster's central parts, compact enough to avoid tidal disruption. These are only two of the examples illustrating the SFH richness of these objects confirmed with our data.