Scattering scanning near-field optical microscopy (s-SNOM) has emerged as a powerful imaging and spectroscopic tool for investigating nanoscale heterogeneities in biology, quantum matter, and electronic and photonic devices. However, many materials are defined by a wide range of fundamental molecular and quantum states at far-infrared (FIR) resonant frequencies currently not accessible by s-SNOM. Here we show ultrabroadband FIR s-SNOM nanoimaging and spectroscopy by combining synchrotron infrared radiation with a novel fast and low-noise copper-doped germanium (Ge:Cu) photoconductive detector. This approach of FIR synchrotron infrared nanospectroscopy (SINS) extends the wavelength range of s-SNOM to 31 μm (320 cm-1, 9.7 THz), exceeding conventional limits by an octave to lower energies. We demonstrate this new nanospectroscopic window by measuring elementary excitations of exemplary functional materials, including surface phonon polariton waves and optical phonons in oxides and layered ultrathin van der Waals materials, skeletal and conformational vibrations in molecular systems, and the highly tunable plasmonic response of graphene.