The most common and most likely answer to this crossword clue is the 8 letter word PERSISTS
P
E
R
S
I
S
T
S
More clues leading to the result PERSISTS
- Goes on forward, keeping relative sweet at first
- After an upset, bug gripping relative endures
- After quiet start, one beset by terrible stress keeps going
- Soldiers on street — sister and agent heading back
- Obstinately continues appended note about cycling nun
- Beset by pains, problem for manual workers goes on
- Rung up about relative tracking earliest of Roman remains
- Endures insolent son stifling sister
- Continues on both sides of personnel lists
- Historically Iran, for the most part, stores no mineral remains
Buddy explains!

The straight part
We know that "Obstinately continues" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case obstinately continues = persists.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 PERSISTS mean?
PERSISTS verb- To go on stubbornly or resolutely.
- To repeat an utterance.
- To continue to exist.
- To continue to be; to remain.
- To cause to persist; make permanent.