The US is unique in this regard in that anyone born inside the US is automatically deemed to be a US citizen. IS citizens can live outside the US for as long as they like, they never lose their citizenship. Rest assured, there are no exit controls of any kind upon departing the US. You the parents will not be "banned" from coming back, if you'll otherwise meet the visitor's visa requirements in the future.
Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.
Yes, a child born in the U.S. is a US citizen, for the rest of her life. There is no penalty to leave the country, at customs. You will have issues returning to the U.S., not because you gave birth, but if you overstay.
The above is intended only as general information, and does not constitute legal advice. You must speak with an attorney to discuss your individual case.
Yes, she will always be a US citizen.
Mr. Shusterman is a former INS Trial Attorney (1976-82) with over 35 years of immigration experience. His response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.