Thank you for your query at DoctorSpring.com
From the injury that you described, it does not seem likely that she has a fracture.
She is not letting you touch her hand, since it is extremely painful, and being a child, she is scared that any kind of movement, or somebody else trying to hold her hand might cause more pain.
So as you already know, you just need to continue with the RICE protocol.
The pain should subside in a couple of days. In case it doesn't and there is severe restriction of movement, then an x ray of the affected wrist will be necessary ro rule out any fracture or ligament sprain. Being only 3 years, she will have lower threshold of pain, but still within 2 days, her activities should be better. I will advise you to observe her activities, and if you notice she is able to move her wrist in the next 2 days, then RICE with advil is sufficient. A painkiller spray can also be helpful, once you take off the bandage.
But please do not wait beyond a couple of days if her pain does not get better and get the x ray done.
Let me know if i can help you further,