The most common and most likely answer to this crossword clue is the 8 letter word SEDATIVE
I've seen this crossword clue in Financial Times
S
E
D
A
T
I
V
E
More clues leading to the result SEDATIVE
- Drug cops out to keep it back
- Two leaders in sensational case calming down
- Dates I’ve put off for agent with a relaxing nature
- Drug case follows that of suicide
- Small case containing drug which will send you to sleep
- She loses heart: case for sleeping pill?
- Avoids holding appeal backing drug
- Vet with ideas about drug
- Drug could have bearing on the case
- Almost understand case for drug
Buddy explains!

The straight part
We know that "which will send you to sleep" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case which will send you to sleep = sedative.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 SEDATIVE mean?
SEDATIVE noun- An agent or drug that sedates, having a calming or soothing effect, or inducing sleep.
- (pharmaceutical effect) Calming, soothing, inducing sleep, tranquilizing