Monolayer structures of Pb on Si(111) attracted recently considerable interest as superconductivity was found in these truly two-dimensional (2D) structures. In this study, we analyzed the electronic surface band structure of the so-called striped incommensurate Pb phase with 43 ML coverage by means of spin-resolved photoemission spectroscopy. Our results fully agree with density functional theory calculations done by Ren et al. [Phys. Rev. B 94, 075436 (2016)1098-012110.1103/PhysRevB.94.075436]. We observe a local Zeeman-type splitting of a fully occupied and spin-polarized surface band at the K̄3 points. The growth of this densely packed Pb structure results in the formation of imbalanced rotational domains, which triggered the detection of C3v symmetry forbidden spin components for surface states around the Fermi energy. Moreover, the Fermi surface of the metallic surface state of this phase is Rashba spin split and revealed a pronounced warping. However, the 2D nesting vectors are incommensurate with the atomic structure, thus keeping this system rather immune against charge density wave formation and possibly enabling a superconducting behavior. © 2017 American Physical Society.