The most common and most likely answer to this crossword clue is the 8 letter word CRIMINAL
I've seen this crossword clue in The Daily Telegraph
C
R
I
M
I
N
A
L
More clues leading to the result CRIMINAL
- Critical, having short period for twitching that's very bad
- Deplorable failing in aim to access credit line
- Thief taking new Mini Cooper emptied out nearly everything
- Credit setter’s given in Alabama is deplorable
- Felon with power abused imperial crown
- Offender in crashed minicar, learner driver
- Villain has border home occupying West Coast state
- Lawless section of Pacific Rim, in Alaska
- Lawless US state keeping border lit
- Deplorable to get caught at boundary, covers are off for last match
Buddy explains!

The straight part
We know that "Offender" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case offender = criminal.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 CRIMINAL mean?
CRIMINAL noun- A person who is guilty of a crime, notably breaking the law.
- Against the law; forbidden by law.
- Guilty of breaking the law.
- Of or relating to crime or penal law.
- Abhorrent or very undesirable.