The hypothesis of a universal initial mass function (IMF) - motivated by observations in nearby stellar systems - has been recently challenged by the discovery of a systematic variation of the IMF with the centralvelocity dispersion, σ, of early-type galaxies (ETGs), towards an excessof low-mass stars in high-σ galaxies. This trend has been derived so farfrom integrated spectra, and remains unexplained at present. To testwhether such trend depends on the local properties within a galaxy, we have obtained new, extremely deep, spectroscopic data, for three nearby ETGs, two galaxies with high σ (~300 km/s), and one lower mass system, with σ ~100 km/s. From the analysis of IMF-sensitive spectral features, we find that the IMF depends significantly ongalactocentric distance in the massive ETGs, with the enhanced fraction of low-mass stars mostly confined to their central regions. In contrast, the low-σ galaxy does not show any significant radial gradient in the IMF, well described by a shallower distribution, relative to the innermost regions of massive galaxies, at all radii. Such a result indicates that the IMF should be regarded as a local (rather than global) property, and suggests a significant difference between the formation process of the core and the outer regions ofmassive ETGs