If you mention the petty stuff, I'd frame it this way: A has made it clear that s/he is unhappy with the work environment. (Do not present it as a personal problem, "A dislikes me.") She's come to you directly with some complaints and other complaints you've only heard secondhand, but still you've done your best to accommodate her, although she's made it clear that it isn't enough. Ask the supervisor directly, "Is there anything you think I should be doing differently or something I could change that would help A work better in our group?" This is a very important question, since it will show the supervisor that you want to fix this, not just complain. The answer will give you some insight into the supervisor's perception of you, and s/he just might have some good ideas or information about A that you don't have. Be sincere about asking and answering. This is productive problem-solving, not complaining.
Thanks, I will remember it for Monday, when I finally have that meeting with the supervisor.
As I type I wonder if one of the problems here might be that the chain of responsibility is not clear. Are you in a situation where you are responsible for telling her what to do but have no real authority in evaluating her or hiring/ firing? Is it possible that you and she have different perceptions of your work relationship and she feels that you are equals?
Absolutely, I am supposed to guide A, but I can do nothing at all if A refuses. The last exchange with A suggests that A feels superior actually, I was "told off" (in front of colleagues) for my project contribution. I was so in shock, I did not say anything as I had absolutely no idea what to say. Supervisor was also present, did not say much yet, but that definitely merits discussions on Monday.
And I wonder too, how do your other colleagues perceive this situation? Do they think that A is in the wrong, or that A is being mistreated, or what? I think you need to let your supervisor be aware of your side, since otherwise A may sound credible. But you need to do it in a way that focuses on the real problem (success of project) and does not entail direct complaining.
Some colleagues are not aware of issues (I do not gossip), some find A weird. It is a slow realization since A is work-wise linked to me, and others only see the non-work side of A.
One thing I'm guessing about you: if you are considering leaving a job because one person junior to you is being so unpleasant, without even speaking up, there is something else going on here. Are you the kind of person who just shuts down when there is any conflict, paired with a colleague who apparently thrives on conflict? Are you working in a foreign environment or one where you somehow feel like an outsider?
I am lucky to work in a field with plenty of jobs (even now). There is no need for me to continue working in an environment where it gets to the point that I dislike coming to work so much. So far I have never experienced such conflicts, I thought of myself as pretty easy -going in that respect.