In many unconventional superconductors, the presence of a pseudogap - a suppression in the electronic density of states extending above the critical temperature - has been a long-standing mystery. Here, we employ combined in situ electrical transport and angle-resolved photoemission spectroscopy measurements to reveal an unprecedentedly large pseudogap regime in single-layer FeSe/SrTiO3, an interfacial superconductor where incoherent Cooper pairs are initially formed above TΔ≈60 K but where a zero-resistance state is achieved only below T0<30 K. We show that this behavior is accompanied by distinct transport signatures of two-dimensional phase fluctuating superconductivity, suggesting a mixed vortex state hosting incoherent Cooper pairs which persist well above the maximum clean limit Tc of approximately 40 K. Our work establishes the critical role of reduced dimensionality in driving the complex interplay between Cooper pairing and phase coherence in two-dimensional high-Tc superconductors, providing a paradigm for understanding and engineering higher-Tc interfacial superconductors. © 2021 authors. Published by the American Physical Society.