The most common and most likely answer to this crossword clue is the 7 letter word SNOOZES
I've seen this crossword clue in The Guardian
S
N
O
O
Z
E
S
More clues leading to the result SNOOZES
Buddy explains!

The straight part
We know that "is dropping off" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case is dropping off = snoozes.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 SNOOZES mean?
SNOOZES noun- A brief period of sleep; a nap.
- The snooze button on an alarm clock.
- Something boring.
- To sleep, especially briefly; to nap, doze.
- To pause; to postpone for a short while.