The low-energy excitations of the lightly doped cuprates were studied by angle-resolved photoemission spectroscopy. A finite gap was measured over the entire Brillouin zone, including along the d_{x^2 - y^2} nodal line. This effect was observed to be generic to the normal states of numerous cuprates, including hole-doped La_{2-x}Sr_{x}CuO_{4} and Ca_{2-x}Na_{x}CuO_{2}Cl_{2} and electron-doped Nd_{2-x}Ce_{x}CuO_{4}. In all compounds, the gap appears to close with increasing carrier doping. We consider various scenarios to explain our results, including the possible effects of chemical disorder, electronic inhomogeneity, and a competing phase.