The ⁸⁷Rb-⁸⁷Sr radiochronometer provides key insights into the timing of volatile element depletion in planetary bodies, yet the unknown nucleosynthetic origin of Sr anomalies in Ca-Al-rich inclusions (CAIs, the oldest dated solar system solids) challenges the reliability of resulting chronological interpretations. To identify the nature of these Sr anomalies, we performed step-leaching experiments on nine unmelted CAIs from Allende. In six CAIs, the chemically resistant residues (0.06 to 9.7% total CAI Sr) show extreme positive μ ⁸⁴Sr (up to +80,655) and 87Sr variations that cannot be explained by decay of ⁸⁷Rb. The extreme ⁸⁴Sr but more subdued ⁸⁷Sr anomalies are best explained by the presence of a presolar carrier enriched in the p-nuclide ⁸⁴Sr. We argue that this unidentified carrier controls the isotopic anomalies in bulk CAIs and outer solar system materials, which reinstates the chronological significance of differences in initial ⁸⁷Sr/⁸⁶Sr between CAIs and volatile-depleted inner solar system materials.