# Does any uncountable complete theory have exactly two countable models?

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.