We present the first resolved study of the radio continuum properties of I Zw 18, the dwarf galaxy with the lowest known nebular metal abundance in the local universe. New Very Large Array radio continuum images at 20 and 3.6 cm are compared to various Hubble Space Telescope images, and we find a striking morphological similarity between high-resolution Hα and short-wavelength radio continuum emission, especially in the Hα shell in the northwest region. We separate thermal and nonthermal components of the emission and find a large synchrotron halo surrounding the galaxy. Comparison between Hα and X-band fluxes suggests that the emission at 3.6 cm is dominated by thermal processes; an additional synchrotron component dominates the flux at 20 cm and produces a modest fraction of the detected flux at 3.6 cm. The fluxes of three of the four major emission peaks show a mix of thermal and nonthermal processes, while one shows a nearly flat spectral index. The strong synchrotron component argues for active star formation throughout the disk for at least the last ~30 Myr. These sensitive observations provide a new, detailed view of the nature of radio continuum emission in the very low metallicity interstellar medium. Comparing with the literature, the role of metallicity in the evolution of radio continuum emission seems to be secondary to other factors such as the recent star formation history and the presence or absence of outflows from star formation regions.