Beryllium-10 is a short-lived radionuclide (t_1/2_=1.4Myr) uniquely synthesized by spallation reactions and inferred to have been present when the solar system's oldest solids (calcium-aluminum-rich inclusions, CAIs) formed. Yet, the astrophysical site of ^10^Be nucleosynthesis is uncertain. We report Li-Be-B isotope measurements of CAIs from CV chondrites, including CAIs that formed with the canonical ^26^Al/^27^Al ratio of ~5x10^-5^ (canonical CAIs) and CAIs with Fractionation and Unidentified Nuclear isotope effects (FUN-CAIs) characterized by ^26^Al/^27^Al ratios much lower than the canonical value. Our measurements demonstrate the presence of four distinct fossil ^10^Be/^9^Be isochrons, lower in the FUN-CAIs than in the canonical CAIs, and variable within these classes. Given that FUN-CAI precursors escaped evaporation-recondensation prior to evaporative melting, we suggest that the ^10^Be/^9^Be ratio recorded by FUN-CAIs represents a baseline level present in presolar material inherited from the protosolar molecular cloud, generated via enhanced trapping of galactic cosmic rays. The higher and possibly variable apparent ^10^Be/^9^Be ratios of canonical CAIs reflect additional spallogenesis, either in the gaseous CAI-forming reservoir, or in the inclusions themselves: this indicates at least two nucleosynthetic sources of ^10^Be in the early solar system. The most promising locale for ^10^Be synthesis is close to the proto-Sun during its early mass-accreting stages, as these are thought to coincide with periods of intense particle irradiation occurring on timescales significantly shorter than the formation interval of canonical CAIs.