Hi, B.. I'm a mom to a ballet dancer (nearly 13) who has danced since she was about three -- that's just so you know where I'm coming from.
I think that your daughter was pretty honest and self-aware to be able to express that she really is upset about missing play time, more than she is disliking ballet itself. Did you talk to her about "What DO you like in ballet class?" See if you can draw her out to talk about whatever positives she finds there -- maybe she likes the teacher, has friends there, likes certain things they do even if she doesn't like other things (for instance, she might really like the result once they learn a full dance but doesn't really enjoy barre work). That might give her a little perspective that ballet isn't really as bad as she thinks; she would miss her friends there if she quits, etc. But don't force it -- let her be the one to think through the positives.
Also, is she in class only once a week? That actually is not that much time. How much play time would she really gain if she gave it up? Maybe two hours total if you include getting there and back? Something else to talk to her about.
Is it a formal and strict sort of class or is there at least some fun? A lot of this depends on the teacher and how "serious" the studio is. Any decent class of course has to teach the basics but a class for kids her age should have some elements of fun in it. Have you observed it a lot?
Then there is the commitment. If you are signed up for a class that runs Sept. to June, it would be a natural break to pull her out after the holidays; the studio is probably going to have other kids who quit then, and it won't be a big deal on their end or yours - unless you have pre-paid for a whole year, in which case you need to talk to the studio. I am NOT in favor of kids grinding along doing any activity that they truly hate, IF they have at least given it a real chance first; if she has been taking ballet since school started, she has been doing it long enough to know what's up, so....
Have you considered if she would like another dance form better? Does the studio offer other types of dance? I would really try to find a shorter-term class in jazz or modern or lyrical or whatever -- not a full-year class but a six- or 10-week class, possibly at a recreation center if not a studio. That way she gets a taste but is not locked in for many months, and you can end it naturally when the class period is done, or continue, or find a longer-term class at a studio.
One very important thing, B. -- you mention late in the post that you "did ballet my whole life." I think you know what I"m about to say, right? You loved it but she might not, and please don't let your own desire to see her dance, like you did, influence you if she really is not right for ballet and it's not right for her. She may quit now and then want to try again later. Or she may quit now but want to do a summer-only class in ballet (or, again, another dance form). Be careful that your own passions and past don't influence you too much. I know that's hard but you can do it.
Do you take her to ballets (including performances by other schools and studios)? Does she like to watch dance and ballet videos and dance along at home? If she does not -- that may be a sign she's just not into dance right now. Don't force it, whatever you do. If you were the one suggesting ballet, and she stops, tell her that she gets to pick the next activity (with you having veto power of course!) and work with her to try something utterly diffferent -- kids' drama, art, an individual sport (we know a girl who's been fencing since she was six) or whatever. Just do shorter-term classes, enough to get a good taste but not a nine-month commitment.
Some of her resistance may be just normal six-year-old resistance to "having" to do anything other than hang out and play. I do believe in activities for kids, within reason, and dance has been huge for us, but your daughter might like to try several things. She's still young -- yes, many dancers start very young but others start much older and do fine. I know kids who started at eight or nine or 10 and they "caught up" well because they were mature enough to deal with the repetition of ballet and see the end goal.