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 central velocity dispersion, σ, of early-type galaxies (ETGs), towards an excess of low-mass stars in high-σ galaxies. This trend has been derived so far from integrated spectra, and remains unexplained at present. To test whether 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- 1), and one lower mass system, with σ ˜ 100 km s- 1. From the analysis of IMF-sensitive spectral features, we find that the IMF depends significantly on galactocentric 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 of massive ETGs.