The most common and most likely answer to this crossword clue is the 9 letter word DROWSIEST
I've seen this crossword clue in Financial Times
D
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More clues leading to the result DROWSIEST
Buddy explains!

The straight part
We know that "Extremely sleepy" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case extremely sleepy = drowsiest.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 DROWSIEST mean?
DROWSIEST adjective- Inclined to drowse; heavy with sleepiness
- Causing someone to fall sleep or feel sleepy; lulling; soporific.
- Boring.
- Dull; stupid.