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The overdoped side of the cuprate phase diagram is generally considered to be the most conventional: a BCS-like superconductor emerging from a Fermi liquid normal state. Surprisingly, it was recently found that the superfluid drops to zero with increased overdoping, breaking with the previous consensus. The origin of this anomalous reduction is unknown_. Our scanning tunneling spectroscopy measurements in the overdoped regime of the (Pb,Bi)₂Sr₂CuO_6+δ high-temperature superconductor show that it is due to the emergence of a heterogeneous state with nanoscale superconducting puddles in a metallic matrix, confirming theoretical proposals within the Bardeen-Cooper-Schrieffer (BCS) mean-field theory. Our measurements further reveal that this puddling is driven by gap filling, not gap closing. Unexpectedly, we find that a basic rule from the mean-field description is dramatically violated: mean-field theory predicts that pair-breaking should always go hand in hand with a diminishment of the gap magnitude, but what we observe instead is that a larger gap coincides with stronger pair breaking. This suggests that the mechanism governing superconductivity is qualitatively different from the conventional mean-field theory.