Hi P.!
Potty time is not play time! When we were training our daughter, if she said she had to go, we would take..EVERY TIME! You have to or she will not tell you when she REALLY has to go...keep that up! The next step is what you need to change.
Once you are there, if she doesn’t go within one minute, get her out of there. Otherwise it will become a game, just like you mentioned; a place to play. Even if you have to go back in there 5 minutes later, don't let her dwell. It is going to be really aggravating for awhile, but you'll get through it. We taught our daughter to “relax” by breathing in and out. We found she was tensing and stopping the flow, so nothing would happen. Once she figured that out, it was easier.
On another note, we tried the potty training thing at age 2 and my daughter wasn’t ready. We stopped for a bit and then re-approached every few months until we got better cooperation. We also experienced her going in the potty early on, but then she did what your little girl is doing (and threw tantrums!), so we cooled it for a bit and re-approached.
If she is going frequently, then keep it up because you are there, but if it is completely hit & miss, maybe a break (which is what my pediatrician recommended) is a good idea. The 3rd time we did that, Emily approached us about the potty and then you KNOW it’s gonna work! :o)
Good luck and hang in there!!!