I wouldn't ever give up on trying, but let the kid lead you to what she needs. When I was about fifteen I completely freaked just as I was starting to learn a RO-BHS. This eventually lead to me being scared to do most backwards things - back tucks off beam, standing BHS on floor, flyaways. At the time I sortof knew that I needed to take more time with the basics and continue to work the skills with a spot. I ended up refusing to do floor (I was on the high school team so it's a lot easier to do this than a club gymnast). Now, 25 years later, I'm starting to tumble again. I'm slowly working back to tumbling with a new awareness of what I need.
I wasn't a very good gymnast, but the fact that I could continue doing the sport while working around these fears means I've got a lifelong commitment to the sport. I've coached rec, high school and middle school, I now judge, I plan to go back to coaching soon. The sport needs coaches and judges so remember your not-so-good gymnasts with major fear issues should still be nurtured to love the sport.
You need to make sure the kid is physically and mentally able to do the skill. Many coaches say "you can do this" and physically the kid can. But if the mental side of things isn't there, it's not going to happen. There's a fine line between pushing a kid through normal fear and pushing against a mental unreadiness. It's really hard to tell the difference. I suspect you'll know when the fear shows up in skills the kid has been doing for a long time or it's moved from one skill to a similar skill on a different event.