Radiation doesn't stay around for more than a millisecond.

Radioactive material can stick to things and keep producing radiation. You want to avoid getting radioactive material on you or your clothes, and washing it off is effective: The material will still produce radiation but now it doesn't do that on your skin (or worse, in your body if you inhale o eat parts of it). Typically cleaning is much faster than the half life of the substance being handled.

To a really good approximation, radiation doesn't make other things radioactive. Induced radioactivity is an extremely weak effect. It's only relevant if you remove material from a nuclear reactor or parts of particle accelerators where they have been exposed to very high radiation levels.