We report on a study of epitaxially grown ultrathin Pb films that are only a few atoms thick and have parallel critical magnetic fields much higher than the expected limit set by the interaction of electron spins with a magnetic field, that is, the Clogston-Chandrasekhar limit. The epitaxial thin films are classified as dirty-limit superconductors because their mean-free paths, which are limited by surface scattering, are smaller than their superconducting coherence lengths. The uniformity of superconductivity in these thin films is established by comparing scanning tunneling spectroscopy, scanning superconducting quantum interference device (SQUID) magnetometry, double-coil mutual inductance, and magneto-transport, data that provide average superfluid rigidity on length scales covering the range from microscopic to macroscopic. We argue that the survival of superconductivity at Zeeman energies much larger than the superconducting gap can be understood only as the consequence of strong spin-orbit coupling that, together with substrate-induced inversionsymmetry breaking, produces spin splitting in the normal-state energy bands that is much larger than the superconductor's energy gap.superconductivity is a topic of growing interest in contemporary condensed matter physics. Early experimental work in this field used granular thin films to study phase transitions to insulating normal states driven by weakened superfluid rigidity in the ultrathin film regime. Recent experimental progress (1-12) in epitaxial growth of uniform 2D superconductors whose properties are largely intrinsic has opened up new possibilities for the design of superconducting systems with specific desirable physical properties. Indeed, these almost ideal 2D systems have yielded (6, 12, 13) surprisingly robust superconductivity in films that are only one or two atomic layers thick, and very recently the observation of an astonishingly high T c in singlelayer FeSe on SrTiO 3 (14-17). In addition, because 2D superconductors must be placed on a substrate, they necessarily have broken inversion symmetry and Rashba-type spin-orbit interactions that break the spin degeneracy of quasiparticle levels in the normal state, and enable the possibility of achieving topological (18-21) superconducting states.Here, we investigate the superconducting properties of strong spin-orbit coupling 2D superconductors using epitaxially grown, ultrathin Pb films on Si. By establishing that the superfluid rigidity vanishes at essentially the same T c when measured on different length scales, from atomic to macroscopic (greater than millimeter), we demonstrate the uniformity of the superconductivity in our films and obtain highly reliable quantitative superfluid density (SFD) values. We then perform magneto-transport measurements in parallel fields, which clearly establish that the Clogston-Chandrasekhar (CC) limit does not apply to our films. Superconductivity at Zeeman fields well in excess of the superconducting energy gap can be understood only as a consequence of strong...