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The polarization of the Lambda and (Lambda) over bar hyperons along the beam direction has been measured in proton-lead (p-Pb) collisions at a center-of-mass energy per nucleon pair of 8.16 TeV. The data were obtained with the CMS detector at the LHC and correspond to an integrated luminosity of 186.0 +/- 6.5 nb(-1). A significant azimuthal dependence of the hyperon polarization, characterized by the second-order Fourier sine coefficient P-z,P-s2, is observed. The P-z,P-s2 values decrease as a function of charged particle multiplicity, but increase with transverse momentum. A hydrodynamic model that describes the observed P-z,P-s2 values in nucleus-nucleus collisions by introducing vorticity effects does not reproduce either the sign or the magnitude of the p-Pb results. These observations pose a challenge to the current theoretical implementation of spin polarization in heavy ion collisions and offer new insights into the origin of spin polarization in hadronic collisions at LHC energies.