The most common and most likely answer to this crossword clue is the 3 letter word BIG
I've seen this crossword clue in The Guardian
B
I
G
More clues leading to the result BIG
- Sizeable British force on the ground
- Bishop, one old senior servant, from the south to give approval
- This shot an important person
- Important American soldier returned after onset of battle
- Rock climbing is great
- Book on island, good, important
- Generous portion of grub I got
- Generous wedge served up
- & 19dn. Vast whorehouse almost at end of tether, as seen on TV
- Being regularly culled is important
Buddy explains!

The straight part
We know that "important" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case important = big.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 BIG mean?
BIG noun- Someone or something that is large in stature
- An important or powerful person; a celebrity; a big name.
- (as plural) The big leagues, big time.
- (BDSM) The participant in ageplay who acts out the older role.
- To praise, recommend, or promote.
- Of great size, large.
- (of an industry or other field, often capitalized) Thought to have undue influence.
- Popular.
- Adult.
- Fat.
- Important or significant.
- (with on) Enthusiastic (about).
- Mature, conscientious, principled; generous.
- Well-endowed, possessing large breasts in the case of a woman or a large penis in the case of a man.
- (sometimes figurative) Large with young; pregnant; swelling; ready to give birth or produce.
- Used as an intensifier, especially of negative-valence nouns
- (of a city) populous
- (of somebody's age) old, mature. Used to imply that somebody is too old for something, or acting immaturely.
- In a loud manner.
- In a boasting manner.
- In a large amount or to a large extent.
- On a large scale, expansively.
- Hard.
- To inhabit; occupy
- To locate oneself
- To build; erect; fashion
- To dwell; have a dwelling
- One or more kinds of barley, especially six-rowed barley.