We define p-adic BPS or pBPS invariants for moduli spaces M-beta,M-chi of one-dimensional sheaves on del Pezzo and K3 surfaces by means of integration over a non-archimedean local field F. Our definition relies on a canonical measure mu can on the F-analytic manifold associated to M-beta,M-chi and the pBPS invariants are integrals of natural G(m) gerbes with respect to mu(can). A similar construction can be done for meromorphic and usual Higgs bundles on a curve. Our main theorem is a chi-independence result for these pBPS invariants. For one-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of pBPS with usual BPS invariants through a result of Maulik and Shen [Cohomological chi-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles, Geom. Topol. 27 (2023), 1539-1586].