I can't find anything here [0][1] which implies that embassies are foreign soil. I believe, for example, that if, hypothetically, a woman were to give birth in a US embassy they would not become a US citizen.
[0] https://legal.un.org/ilc/texts/instruments/english/conventio...
[1] https://fam.state.gov/FAM/08FAM/08FAM030101.html#M301_1_3