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The strong equivalence principle (SEP) distinguishes General Relativity from other viable theories of gravity. The SEP demands that the internal dynamics of a self-gravitating system under free-fall in an external gravitational field should not depend on the external field strength. We test the SEP by investigating the external field effect (EFE) in Milgromian dynamics (MOND), proposed as an alternative to dark matter in interpreting galactic kinematics. We report a detection of this EFE using galaxies from the Spitzer Photometry and Accurate Rotation Curves (SPARC) sample together with estimates of the large-scale external gravitational field from an all-sky galaxy catalog. Our detection is threefold: (1) the EFE is individually detected at $8σ$ to $11σ$ in "golden" galaxies subjected to exceptionally strong external fields, while it is not detected in exceptionally isolated galaxies, (2) the EFE is statistically detected at more than $4σ$ from a blind test of 153 SPARC rotating galaxies, giving a mean value of the external field consistent with an independent estimate from the galaxies' environments, and (3) we detect a systematic downward trend in the weak gravity part of the radial acceleration relation at the right acceleration predicted by the EFE of the MOND modified gravity. Tidal effects from neighboring galaxies in the $Λ$CDM context are not strong enough to explain these phenomena. They are not predicted by existing $Λ$CDM models of galaxy formation and evolution, adding a new small-scale challenge to the $Λ$CDM paradigm. Our results point to a breakdown of the SEP, supporting modified gravity theories beyond General Relativity.