The most common and most likely answer to this crossword clue is the 7 letter word CHASERS
I've seen this crossword clue in The Times
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Buddy explains!

The straight part
We know that "People pursuing/further drinks" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case people pursuing/further drinks = chasers.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 CHASERS mean?
CHASERS noun- A person or thing (ship, plane, car, etc.) that chases.
- A hunter.
- A person who does the chasing on metalwork.
- A horse: a horse used for hunting; a horse trained for steeplechasing, a steeplechaser.
- A drink drunk after another of a different kind
- (logging) Someone that follows logs out of the forest in order to signal a yarder engineer to stop them if they become fouled (also called a frogger).
- (logging) One who unhooks chokers from the logs at the landing.
- A piece of music, etc. played after a performance while the audience leaves.
- One of a series of adjacent light bulbs that cycle on and off to give the illusion of movement.
- A person who seeks out sexual partners with a particular quality:
- Someone who chases (decorates) metal; a person who decorates metal by engraving or embossing.
- A tool used for cleaning out screw threads, either as an integral part of a tap or die to remove waste material produced by the cutting tool, or as a separate tool to repair damaged threads.
- A chase gun.
- A prison guard; particularly one who guards military prisoners on fatigue duty