Father's status known or unknown?
If known and US citizen then probably.
If not known or foreign national or dual citizen....I would say no.
No other country could ever lay claim to child and the child was not 'governed' by other country laws in any way, shape or form what-so-ever.
Try turning the question around. What if a non-American couple had a baby aboard a US warship in international waters? Would the kid be an American by reason of the 14th Amendment? If not, then the ship isn't really US territory, is it? Same logic applies to US bases in foreign lands.
As for your original question, the answer is yes, because of the citizenship of the mother. She being American, we don't have to decide whether the boat is US territory.
That also would apply to Obama, BTW, if he were somehow shown to have been born abroad. He would still be eligible because of his mother. And the oft-mentioned age requirement on the mother would not apply because her marriage was invalid on account of BHO, Sr.'s bigamy.
In order to prove Obama ineligible, you would have to show not only that he was born abroad but also that some floozy from Liverpool, for instance, was his mother, and not Stanley Ann, born in Kansas. And then you'd have to get a majority of the House and 67 senators to agree to oust him. None of that is going to happen.