My son gave up naps around age 3. It wasn't ideal. He just decided he didn't want to nap anymore. I think some of this stemmed from the fact that he saw his baby sister taking naps during the day and decided that naps were for babies. Every so often, like if he's sick, he'll take a nap.
Despite this, though, he still has a daily quiet time. It may involve playing quietly, coloring, looking at books, playing with Legos, but, more often than not, vegging out on the couch and watching an age-appropriate show or two. The deal is that he can't make a lot of noise (so as not to wake up his sister--in fact, he's not even allowed upstairs while she's sleeping), and he can't disturb us (Mom and Dad) as we frequently use that time to clean, check email, make phone calls, etc.
We've also rolled back his bedtime to ensure that he's getting plenty of sleep at night. He was becoming insanely cranky as the evening wore on and we finally explained to him that if he wasn't going to take a nap then he HAD to go to bed earlier to make sure he was getting enough sleep. He didn't like it at first, but he's settled in to the routine and generally falls asleep very quickly at night now (because he's exhausted!).
The other thing to consider is that this might just be a temporary power-struggle thing. She's 3--it's entirely possible that she's just flexing her 3-year-old muscles to see what she can get away with. I had a friend of mine whose daughter tried the same thing around age 3. She eventually went back to taking naps again a few months later on her own. In the meantime, I think if you try to force it, your daughter will just try to fight harder to avoid the nap. Giving her a quiet place (it doesn't have to be her bedroom) where she can sit alone and do something quietly until naptime is over might be your best option.
