The idea that radiation turns materials radioactive is, for the most part, false. Of all the commonly encountered types of radiation (electromagnetic including gamma, alpha, beta, neutron), only neutron radiation is capable of making other things radioactive.
Any material exposed to the other kinds will never develop its own radioactivity, however if radioactive dust sticks to the material then this dust will of course continue to be radioactive and make your shielding radioactive until it is cleaned.
For neutron radiation specifically, some cursory research indicates that lead exposed to neutron radiation does become somewhat radioactive, although it would take a lot of neutrons to make it super radioactive. It appears that it would radiate mostly beta radiation once made radioactive, although possibly some gamma as well.
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