The Short Version
This claim is true. Bananas contain potassium-40 (K-40), a naturally occurring radioactive isotope that makes up about 0.012% of all potassium. This is confirmed by the US EPA, the Department of Energy, and peer-reviewed scientific literature. However, the radioactivity is extremely small — about 0.1 microsieverts per banana — and eating bananas does not increase your net radiation dose because the body maintains potassium balance and excretes excess potassium. Bananas pose no radiation health risk.