Strengthen Lowenheim-Skolem

Question

Suppose $\frak A \prec \frak B$ are models in a language $\mathcal L$, and $|\frak A| < \kappa < |\mathcal L| \leq |\frak B|$. Is there an elementary chain $\frak A \prec \frak C \prec \frak B$ such that $|\frak C| = \kappa$?

Answer 1

It is at least consistent that the answer is "no". This answer of mine gives an example of a theory in a continuum-sized language with a countable model $\mathfrak{A}$ such that any proper elementary extension is of size at least continuum. If $2^{\aleph_0}>\aleph_1$, then taking $\kappa=\aleph_1$ gives a counterexample to the question.