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 4/3 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)]. We observe a local Zeeman-type splitting of a fully occupied and spin-polarized surface band at the band at the (K) over bar root 3 points. The growth of this densely packed Pb structure results in the formation of imbalanced rotational domains, which triggered the detection of C-3v 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.