Ada answered the question, but consider your question.

You said “leaving a spaceship with no suit”. If your idea is that something left in space becomes dangerously radioactive then any space station or space ship would itself become dangerously radioactive.

So the answer is, no, things in space don’t become dangerously radioactive. Also things in contact with radioactive substances don’t themselves become radioactive except under extremely specific circumstances. Your house didn’t become dangerously radioactive because of the radioactive americium-241 in the smoke detectors.