The most common and most likely answer to this crossword clue is the 8 letter word SECEDING
I've seen this crossword clue in Financial Times
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Buddy explains!
The straight part
We know that "going one’s own way" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case going one’s own way = seceding.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 SECEDING mean?
SECEDING verb- To split from or to withdraw from membership of a political union, an alliance or an organisation.
- To split or to withdraw one or more constituent entities from membership of a political union, an alliance or an organisation.