As the Parker Solar Probe explores new regions of the inner heliosphere, it travels ever deeper into the electric potential of the Sun. In the near-Sun environment, a new feature of the electron distribution emerges, in the form of a deficit in the sunward suprathermal population. The lower boundary of this deficit forms a cutoff in phase space, at an energy determined by the electric potential drop between the observation point and the outer heliosphere. We explore the characteristics of the sunward deficit and the associated cutoff, as well as the properties of the plasma in which we observe them. The deficit occurs in ~60%-80% of electron observations within ~0.2 au, and even more frequently in plasma with low β, low collisional age, and a more anisotropic electron core population. At greater distances, the deficit rapidly disappears, as the suprathermal halo grows, with these two trends likely related. The cutoff energy varies linearly with the local electron core temperature, confirming a direct relationship to the ambipolar electric potential. Meanwhile, the cutoff width varies with β and collisional age, suggesting that energy diffusion plays a role in erasing the deficit. The nearly ubiquitous occurrence of the sunward deficit in the inner heliosphere suggests that we may need to reconsider the functional forms commonly used to represent electron distributions in this environment.