To me, this is so dependent on "know your own kid / know the history and patterns of behavior" to give any specific advice. For example...
Is she a kid that frequently gets frustrated and 'quits' things when they become a little hard? Or a kid that never gives up on anything typically?
Is it actually a skill (or physical challenge such as harder conditioning) triggering this? For how long? And what has been the approach by the coach? Do you ever have check ins with the coach to monitor patterns?
How does she typically handle a suddenly tough challenge after a period of relative easy success? Or is this the first time she is facing something truly 'hard and scary' for her?
Was she formerly a gung-ho, gym-is-my-life, and now a sudden change, or has she always been more laid back, more casual gymmie from the start as you look back?
What are the situations at school or in her friends circle that might be affecting this at this moment (sleepovers she is now missing, deeper friendships forming at this age - I have an 8 year old DD, too
)....
As I type this, I see you commented about the BHS on beam, and the threat to move down a level affecting her confidence.
With your new information (and also considering the above personality and experience factors), it sounds like she is facing the common fear of disappointing others, along with the test of her own self-efficacy for gymnastics as the skills are starting to get harder. Personally, I would try my best to use this experience as a teaching moment about sticking with things when they get hard, and not letting the first tough skill define her "success" as a gymnast and enjoyment of gymnastics. And (depending on your real assessment of the coach's style) pointing out how the coach's style in 'pushing her' is intended to raise her confidence and belief in herself, but that it's not always the most effective approach for everyone. Assure her the coach does indeed believe in her, though, and that's why s/he is pushing her hard to do her best. And I always throw in that she could never disappoint me, now matter how long it takes to get a skill, or whatever score comes up on a card, and I'll always support her. I'd encourage her as best I could to keep giving this a go, as it is completely normal to encounter fears and tough skills. Of course, none of it might work, and she still might quit (which is fine), but these lessons are (imo) so key to life in general.
I remember personally be lauded with lots of high-praise labels for my early successes at various things - math, spelling bee, art, track ... And when it started to be a little hard as I got older and there was a threat to me not automatically being the top student any more in these events, I panicked and dropped out of these activities I loved - mainly for fear of disappointing the people around me. I wish my parents had worked a little harder to teach me that these feelings are normal, and it's OK not to always just do things that come easily - maybe I would have stuck with them - even if I wasn't always #1. So my response is biased from my past (as we all are, of course)...