Wow, I had no idea this was so common. This happened to my daughter---twice. The first time, I took her to urgent care because it was a Saturday, and I was sure it was in her ear lobe. The doctor treated me like I was nuts, and sent me home. Well, by Monday her earlobe was more swollen, and when I pressed on it, you could see the butterfly shape of the earring back. It had to be surgically removed (in the Dr's office, but it took maybe 20 minutes--it wasn't coming out). About a year later, she decided to try again . . . same thing happened.
I've let DD get a couple of other piercings, but only from a real piercing place (no more ear guns in the mall). She's gotten her nose and her eyebrow pierced with no problem. Still, her experiences with her ear piercings have made me say no no no to getting her tongue pierced (could you imagine? I will not be responsible for that happening, even though I think the day she turns 18 she may get it done).
OP--I would take your daughter to the doctor asap. If it's grown in there like my DD's, it can't be just pushed out. It wasn't a fun "surgery," but it isn't major, either. You can hold her hand the whole time.
DD had a local anesthetic and there was very little scarring (one reason it took a while was the doctor was very careful to remove it in a way that left no mark on the front, and a minimal one on the back of the lobe). Good luck, and btw your daughter may very well have a metal allergy. If your DD ever has her ears pierced again, it might be better to go to a place that uses a hollow needle that cores out a sliver of ear lobe. Ear guns push the earring in, but there's still as much earlobe there, so there's more pressure. (Maybe I'm assuming, but the eargun/metal allergy combination is what caused my DD's problem.)