The most common and most likely answer to this crossword clue is the 9 letter word INSISTENT
I've seen this crossword clue in The Sunday Telegraph
I
N
S
I
S
T
E
N
T
More clues leading to the result INSISTENT
- Urgent aim to protect little sibling
- Time in playing tennis is demanding
- Pressing aim to welcome relative
- Playing tennis is, at first, terribly demanding
- Urgent plan to capture relative
- Forceful in relation to every new thing, on all fronts
- Concentrating on capturing one on board, compelling attention
- Isn’t twice wrong about drug, not accepting negative response
- Muggins is tentatively keeping resolute
- Determined to include relative in plan
Buddy explains!

The straight part
We know that "not accepting negative response" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case not accepting negative response = insistent.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 INSISTENT mean?
INSISTENT adjective- Standing or resting on something.
- Urgent in dwelling upon anything; persistent in urging or maintaining.
- Extorting attention or notice; coercively staring or prominent; vivid; intense.
- Standing on end: specifically said of the hind toe of a bird when its base is inserted so high on the shank that only its tip touches the ground: correlated with incumbent.