The most common and most likely answer to this crossword clue is the 6 letter word NAVAJO
I've seen this crossword clue in Financial Times
N
A
V
A
J
O
More clues leading to the result NAVAJO
- Native American’s certainly not consuming coffee – it’s rejected
- Language of denial gripping island in recession
- Native American turned up, working round island
- Brave old islander surfacing
- Native American on Java running amok
- Old American about to drink coffee from the East
- Native American‘s refusal to drink coffee from the south
- Language of old islander is dextrosinistral
- American people taking old islander from East to West
Buddy explains!

The straight part
We know that "Language" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case language = navajo.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.