We identified p-type nanoscale dopant-induced dots that are formed by fluctuations of the dopant atom distribution in thin GaAs p-n multilayers. Their electronic structure and the resulting potential variations were investigated by cross-sectional scanning tunneling microscopy and spectroscopy as a function of the number of dopant atoms within the dot. We find significant changes in the current-voltage characteristics of the dots compared to spatially non-confined material, due to a reduced ability to screen the tip's electric field. This indicates a limited ability to deplete the dots of free holes arising from the presence of confining potentials.