Using recently derived results for one-loop hadronic splitting functions from a nonlocal implementation of chiral effective theory, we study the contributions from pseudoscalar meson loops to flavor asymmetries in the proton. Constraining the parameters of the regulating functions by inclusive production of $n$, $Delta^{++}$, $Lambda$ and $Sigma^{*+}$ baryons in $pp$ collisions, we compute the shape of the light antiquark asymmetry $bar{d}-bar{u}$ in the proton and the strange asymmetry $s-bar{s}$ in the nucleon sea. With these constraints, the magnitude of the $bar{d}-bar{u}$ asymmetry is found to be compatible with that extracted from the Fermilab E866 Drell-Yan measurement, with no indication of a sign change at large values of $x$, and an integrated value in the range $langle bar d-bar u rangle approx 0.09-0.17$. The $s-bar s$ asymmetry is predicted to be positive at $x > 0$, with compensating negative contributions at $x=0$, and an integrated $x$-weighted moment in the range $langle x (s-bar s) rangle approx (0.9-2.5) times 10^{-3}$.
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