Fourier-transform infrared (FTIR) spectroscopy(1) is a widely used analytical tool for chemical identification of inorganic, organic and biomedical materials(2), as well as for exploring conduction phenomena(3). Because of the diffraction limit, however, conventional FTIR cannot be applied for nanoscale imaging. Here we demonstrate a novel FTIR system that allows for infrared-spectroscopic nanoimaging of dielectric properties (nano-FTIR). Based on superfocusing(4,5) of thermal radiation with an infrared antenna(4,6), detection of the scattered light, and strong signal enhancement employing an asymmetric(7) FTIR spectrometer, we improve the spatial resolution of conventional infrared spectroscopy by more than two orders of magnitude. By mapping a semiconductor device, we demonstrate spectroscopic identification of silicon oxides and quantification of the free-carrier concentration in doped Si regions with a spatial resolution better than 100 nm. We envisage nano-FTIR becoming a powerful tool for chemical identification of nanomaterials, as well as for quantitative and contact-free measurement of the local free-carrier concentration and mobility in doped nanostructures.