The most common and most likely answer to this crossword clue is the 6 letter word SENATE
I've seen this crossword clue in The Times
S
E
N
A
T
E
More clues leading to the result SENATE
- Legislative body partly chosen at election
- Name occupying chair with English governing body
- Special educational needs bothered governing body
- Cook seen at assembly
- Governing body had to go after a bit of Japanese dosh
- Some chosen at Exeter for the governing body
- Worsen a tendency to expose politicians
- Seat lost and space taken up by government
- Go out with accommodating girl briefly for body
- Ancient Roman assembly depicted in a tense novel
Buddy explains!

The straight part
We know that "governing body" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case governing body = senate.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 SENATE mean?
SENATE noun- In some bicameral legislative systems, the upper house or chamber.
- A group of experienced, respected, wise individuals serving as decision makers or advisors in a political system or in institutional governance, as in a university, and traditionally of advanced age and male.