The following is a theorem by Vaught.
Theorem. Let $T$ be a complete theory in a countable language. Then, $T$ cannot have exactly two countably infinite models (up to isomorphism).
A proof can be found at [Tent-Ziegler, A Course in Model Theory, Theorem 4.3.10].
Does the theorem still hold in an uncountable language? That is:
Question. Let $T$ be a complete theory in an uncountable language. Can $T$ have exactly two countably infinite models?
Any help would be appreciated. Thank you.