Electromagnetic response of individual boron nitride nanotubes (BNNTs) has been studied by spatially resolved electron energy loss spectroscopy (EELS). We demonstrate how dedicated EELS methods using subnanometer electron probes permit the analysis of local dielectric properties of a material on a nanometer scale. The continuum dielectric model has been used to analyze the low-loss EEL spectra recorded from these tubes. Using this model, we demonstrate the weak influence of the out-of-plane contribution to the dielectric response of BNNTs. The optical gap, which can be deduced from the measurements, is found to be equal to 5.8 ± 0.2 eV, which is close to that of the hexagonal boron nitride. This value is found to be independent of the nanotubes configuration (diameter, helicity, number of walls, and interaction between the different walls).