The most common and most likely answer to this crossword clue is the 9 letter word OBSTINATE
I've seen this crossword clue in Financial Times
O
B
S
T
I
N
A
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E
More clues leading to the result OBSTINATE
- Persistent girl stuffing shelled crustacean
- A bit set on changing — or maybe not!
- Unruly Tibetan, so pig-headed!
- Stubborn stain to be specially treated
- Stubborn old boys at home in gallery
- Persistent noise bat produces over time
- Stubborn botanist repositioned bottom of vine
- Stubborn stain needs to be shifted
- Nation defending imperial measure after old boy becomes stubborn
- Stubborn old boy popular in Express
Buddy explains!

The straight part
We know that "Persistent" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case persistent = obstinate.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 OBSTINATE mean?
OBSTINATE adjective- Stubbornly adhering to an opinion, purpose, or course, usually with implied unreasonableness; persistent.
- Said of inanimate things not easily subdued or removed.