Hi there. Since your child was born here she is an American citizen. Unlike is the case for a naturalized citizen or green card holder, nobody can strip her of her citizenship if she acquired it by birth. She can herself formally renounce her citizenship or give it up by becoming a citizen of another country that has no dual citizenship agreement with the U.S. or which contradicts US citizenship, but merely by remaining out of the country will not take away her citizenship. That being said, it is of the utmost importance for you to always carry the proof of her US citizenship in the form of a valid passport. You should be able to renew her passport at the American consulate nearest to the place you will be living abroad.
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Citizenship once acquired by birth or naturalization cannot be lost (rather denounced). As far as renewing passport - can renew at US embassy/consulate abroad.
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This is NOT an immigration law question.
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You written something very peculiar such as" If I decide to take her abroad with me", however, legally you cannot just bluntly take a U.S. born citizen minor abroad for any period of time, especially of a planned prolonged nature you indicated, without an express written authorization of the other parent for that US born child, otherwise your actions will constitute a federal crime of kidnapping.
If in fact your family jointly plans to reside abroad together the U.S. born citizen minor could always renew his/her U.S. Passport at a closest U.S. Consular Section.
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