Cancer is kind of associated with long life span in general, even without external radiation flux, but there are other ways that radiation can be harmful. A given amount of radiation received is more likely to damage cells that quickly divide (especially gametes) and cells that have little or no provision for repair.

So at relatively high levels of radiation you could well see that individual organisms tend to do well enough but have children with more birth defects, are more frail, have their vision deteriorate faster, etc.

I'm not sure that the radiation levels even in the Chernobyl exclusion zone are high enough for that, necessarily, but there's unfortunately a pretty wide gulf between "no effect" and developing cancer.