The most common and most likely answer to this crossword clue is the 9 letter word ANCHORITE
I've seen this crossword clue in Financial Times
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More clues leading to the result ANCHORITE
- Holy recluse has to make fast shortly before mass?
- Her action criminal, she’s in solitary
- Newsreader barely referred to person out of the public eye
- One fluked it, meeting European recluse
- Religious type excited by antechoir
- Short piece in newspaper under “TV presenter becomes religious recluse”
- Hermit as a result of her action, sadly
- Hermit‘s answer – it must be found in new task
Buddy explains!

The straight part
We know that "person out of the public eye" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case person out of the public eye = anchorite.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.
What does ANCHORITE mean?
ANCHORITE noun- One who lives in isolation or seclusion, especially for religious reasons.