The most common and most likely answer to this crossword clue is the 7 letter word ENDEMIC
I've seen this crossword clue in The Times
E
N
D
E
M
I
C
More clues leading to the result ENDEMIC
- Either one leaving medicine out in local
- Death tally rising after hospital goes from a local area?
- Tender skinned tailless rodents prevalent locally
- Kind of ailment almost finished off Mike
- Put a stop to rodents turning up in a particular area
- Death tally back with hospital gone from local area
- One leaving medicine out for native
- Prevalent 23 improbably in charge
- Object to experimental mice being present locally
- Out of unmarried men, half caught disease spreading locally
Buddy explains!

The straight part
We know that "local" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case local = endemic.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 ENDEMIC mean?
ENDEMIC noun- An individual or species that is endemic to a region.
- A disease affecting a number of people simultaneously, so as to show a distinct connection with certain localities.
- Native to a particular area or culture; originating where it occurs.
- (especially of plants and animals) Peculiar to a particular area or region; not found in other places.
- (especially of diseases) Prevalent in a particular area or region.