The most common and most likely answer to this crossword clue is the 7 letter word ETERNAL
I've seen this crossword clue in The Times
E
T
E
R
N
A
L
More clues leading to the result ETERNAL
- Continuous let near resort
- Non-resident wanting vote is persistent
- No time for duck to catch seabird? It’s endless
- Relentless non-resident wanting vote
- Eastern bird coming to a lake, not stopping
- Being always superficial, blowing kiss
- Perennial outside’s no cross
- Everlasting monologue ultimately learnt inadequately
- Endless ale drunk before and after reported act
- Immortal Oriental bird beside a lake
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 = eternal.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 ETERNAL mean?
ETERNAL noun- One who lives forever; an immortal.
- Lasting forever; unending.
- Existing outside time; as opposed to sempiternal, existing within time but everlastingly
- Exceedingly great or bad; used as an intensifier.