The most common and most likely answer to this crossword clue is the 5 letter word SLEPT
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More clues leading to the result SLEPT
Buddy explains!

The straight part
We know that "Dropped off" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case dropped off = slept.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 SLEPT mean?
SLEPT verb- To rest in a state of reduced consciousness.
- (of a spinning top or yo-yo) To spin on its axis with no other perceptible motion.
- To cause (a spinning top or yo-yo) to spin on its axis with no other perceptible motion.
- To accommodate in beds.
- To be slumbering in (a state).
- To be careless, inattentive, or unconcerned; not to be vigilant; to live thoughtlessly.
- To be dead; to lie in the grave.
- To be, or appear to be, in repose; to be quiet; to be unemployed, unused, or unagitated; to rest; to lie dormant.
- To wait for a period of time without performing any action.
- To place into a state of hibernation.