[en] Mechanical techniques for polishing the inside surface of niobium superconducting radio-frequency (SRF) cavities have been systematically explored. By extending known techniques to fine polishing, mirror-like finishes were produced, with <15 nm RMS (root mean square) roughness over 1 mm² scan area. This is an order of magnitude less than the typical roughness produced by the electropolishing of niobium cavities. The extended mechanical polishing (XMP) process was applied to several SRF cavities which exhibited equator defects that caused quench at <20 MV m⁻¹ and were not improved by further electropolishing. Cavity optical inspection equipment verified the complete removal of these defects, and minor acid processing, which dulled the mirror finish, restored performance of the defective cells to the high gradients and quality factors measured for adjacent cells when tested with other harmonics. This innate repair feature of XMP could be used to increase manufacturing yield. Excellent superconducting properties resulted after initial process optimization, with quality factor Q of 3 × 10¹⁰ and accelerating gradient of 43 MV m⁻¹ being attained for a single-cell TESLA cavity, which are both close to practical limits. Several repaired nine-cell cavities also attained Q > 8 × 10⁹ at 35 MV m⁻¹, which is the specification for the International Linear Collider. Future optimization of the process and pathways for eliminating requirements for acid processing are also discussed. (paper)