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Yes, as you have suspected, that could be a first permanent molar starting to come in. Although the expected age for this is 6, there have been some variations in the, especially in girls. If the gum directly behind her last seen baby tooth is slightly swollen as well, then it is almost sure to be because of a adult tooth coming in.
For now, get a local anesthetic gel from the pharmacy to apply to the sore gum so that she can eat without pain. A X ray of that area will confirm the position of the underlying permanent tooth.
However, please get her ear checked from a Ear specialist as well.
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Good Luck !