Plasmonics provides great promise for nanophotonic applications. However, the high optical losses inherent in metal-based plasmonic systems have limited progress. Thus, it is critical to identify alternative low-loss materials. One alternative is polar dielectrics that support surface phonon polariton (SPhP) modes, where the confinement of infrared light is aided by optical phonons. Using fabricated 6H-silicon carbide nanopillar antenna arrays, we report on the observation of subdiffraction, localized SPhP resonances. They exhibit a dipolar resonance transverse to the nanopillar axis and a monopolar resonance associated with the longitudinal axis dependent upon the SiC substrate. Both exhibit exceptionally narrow linewidths (7-24 cm(-1)), with quality factors of 40-135, which exceed the theoretical limit of plasmonic systems, with extreme subwavelength confinement of (λ(res)3/V(eff))1/3 = 50-200. Under certain conditions, the modes are Raman-active, enabling their study in the visible spectral range. These observations promise to reinvigorate research in SPhP phenomena and their use for nanophotonic applications.