The most common and most likely answer to this crossword clue is the 9 letter word MUNICIPAL
I've seen this crossword clue in The Times
M
U
N
I
C
I
P
A
L
More clues leading to the result MUNICIPAL
- German city with hospital demolished by one friend in local government office?
- One comrade going after European city contracts in local government
- Doctored alumni pic belonging to the council
- City-related, pathetic Ukip claim: “Nigel’s head for a thousand!”
- One friend replaces husband in German city or town
- City, German city, mostly one leading knees-up
- Terribly undiplomatic, releasing rebellious Democrat to community
- Run by city, German one mostly, with one friend
- Regarding city affairs, European city reduced independence, linking to China
- ‘City knocked out in Cup’ — Mail
Buddy explains!

The straight part
We know that "in local government" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case in local government = municipal.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 MUNICIPAL mean?
MUNICIPAL noun- A financial instrument issued by a municipality.
- Of or pertaining to a municipality (a city or a corporation having the right of administering local government).
- Of or pertaining to the internal affairs of a nation.