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Scientific and technological progress depend substantially on the ability to image on the nanoscale. In order to investigate complex, functional, nanoscopic structures like, e.g., semiconductor devices, multilayer optics, or stacks of 2D materials, the imaging techniques not only have to provide images but should also provide quantitative information. We report the material-specific characterization of nanoscopic buried structures with extreme ultraviolet coherence tomography. The method is demonstrated at a laser-driven broadband extreme ultraviolet radiation source, based on high-harmonic generation. We show that, besides nanoscopic axial resolution, the spectral reflectivity of all layers in a sample can be obtained using algorithmic phase reconstruction. This provides localized, spectroscopic, material-specific information of the sample. The method can be applied in, e.g., semiconductor production, lithographic mask inspection, or quality control of multilayer fabrication. Moreover, it paves the way for the investigation of ultrafast nanoscopic effects at functional buried interfaces.