The Peierls instability is a foundational mechanism in condensed matter physics, showing how the electron–phonon interaction can transform a simple metal into an insulating state with a new lattice periodicity. In a one-dimensional (1D) metal the Peierls instability follows from the enhanced electronic response at wavevector Q=2k_F (connecting the Fermi points) producing a Kohn anomaly: a softening of the phonon mode at the same wavevector. Condensation of this mode then generates the tell-tale Peierls periodic lattice distortion and charge-density wave (CDW) that gaps the electronic spectrum. Here we show an analogous instability at the edge of a two-dimensional (2D) superconductor, where a dispersing Andreev bound state (ABS) hosts Bogoliubov Fermi points at ± k_c. The enhanced quasiparticle response at the connecting wavevector Q=2k_c couples directly to pairing-fluctuations and produces a superconducting Kohn anomaly: a softening of a pairing mode at the same wavevector. Condensation of this mode then generates an edge pair-density wave (PDW) that gaps the ABS. We refer to this as a superconducting Peierls instability and identify the boundary quasiparticle structure that enables it. As a concrete realization, we consider a square-lattice extended Hubbard model whose mixed-symmetry s+d+ip state hosts a dispersing ABS with zero-energy crossings at finite edge momenta. Using self-consistent Bogoliubov–de Gennes (BdG) calculations we show that the order parameter develops an edge PDW, with the wavevector set by the superconducting Kohn anomaly, that gaps the ABS crossings. Our results identify a novel mechanism for spontaneous translation-symmetry breaking in a superconductor. As this superconducting Peierls mechanism does not require an extensive zero-energy flat band or topologically protected edge states, it may apply broadly to unconventional superconductors.