A pair density wave (PDW), a superconducting state in which the Cooper pair amplitude varies periodically in space, has long been pursued. Among several candidates, its realization in kagome superconductors without a charge density wave remains unresolved, particularly in the strongly correlated regime where an intrinsic PDW may arise, unlike in the FFLO state driven by an external Zeeman field. We investigate possible pairing states in the large-U Hubbard model on the kagome lattice. When the Fermi surface encloses high-symmetry points, a PDW emerges as an inevitable instability. As the chemical potential is tuned, an additional exotic chiral PDW with intra-unit-cell amplitude modulation arises due to phase frustration among the bond-resolved pairing order parameters, and competes with a conventional uniform pairing. Our results establish an intrinsic symmetry-enforced PDW in the correlated regime and identify kagome systems as a natural platform for long-sought PDW states.