The most common and most likely answer to this crossword clue is the 6 plus 5 plus 6 plus 3 letter word PUBLIC ENEMY NUMBER ONE
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Buddy explains!
The straight part
We know that "Dillinger, say" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case dillinger, say = public enemy number one.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.