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Stress introduces anisotropy in the transport coefficients in materials, affecting diffusion. Using first-principles quantum-mechanical methods for activation barriers of atomic jumps, combined with the extended self-consistent mean-field theory to compute transport coefficients with strain-reduced symmetry, we predict significant stress-induced anisotropy for Si impurity diffusion in nickel. This causes complex spatial- and temperature-dependent fluxes; as an example, the heterogenous strain field of a dislocation creates unusual flow patterns that affect mechanical and segregation behavior.