We experimentally study the real-time susceptibility of trapped-ion quantum systems to small doses of ionizing radiation. We expose an ion-trap apparatus to a variety of α, β, and γ sources and measure the resulting changes in trapped-ion qubit lifetimes, coherence times, gate fidelities, and motional heating rates. We found no quantifiable degradation of ion trap performance in the presence of low-dose radiation sources for any of the measurements performed. This finding is encouraging for the long-term prospects of using ion-based quantum information systems in extreme environments, indicating that much larger doses may be required to induce errors in trapped-ion quantum processors.