Sources of polarized leptons are of significant interest for applications in materials physics, nuclear physics, laboratory astrophysics and particle physics. However, existing methods for generating polarized lepton sources typically require stringent experimental conditions, e.g., high-precision spatiotemporal beam-laser alignment or high-charge electron beam. Here, we propose a novel scheme to generate polarized electron and positron sources by injecting an unpolarized electron beam perpendicularly into a sheath field induced by laser-driven plasma expansion. In this scenario, an ultraintense, ultrashort laser pulse heats electrons in a plasma, establishing a strong asymmetric sheath field behind the target. The field is perpendicular to the externally injected electron beams, initiating the nonlinear Compton scattering and leading to radiative spin polarization of the beam. Three-dimensional particle-in-cell simulations show that an average polarization up to $26.5\%$ can be achieved for the entire electron beam, with $51\%$ for those electrons with the energy $<1.5 \ \rm GeV$. Simultaneously, polarized positrons can be also generated via the nonlinear Breit-Wheeler process, achieving a polarization up to $49\%$ and a total charge $0.45 \ \rm nC$. Especially, the target-normal sheath field also naturally separates the resulting polarized electrons and positrons, facilitating the potential use in practical applications. The scheme is notable for its operational stability and relaxed requirements for the driving laser and electron beam, opening new avenues in various fields.