Can a structure have an onto elementary embedding into an elementary substructure?

Question

Let $|A|=|B|\geq \omega$ be structures and $A \prec B$. Suppose there is some $f: B \rightarrow A$ such that $f$ is an elementary embedding. Is it possible for $f$ to be onto? I suspect the answer is no, but I cannot think of a reason why not. I saw a proof that assumed this, and it bothered me.

Answer 1

Just consider dense linear orders without endpoints (in the signature with just $\le$). It is pretty routine that you have one embedded in the other and also they are isomorphic. (And all embeddings are elementary here, cause of quantifier elimination.)

(Also, more basically, let $A=B$ and $f$ be the identity... are you missing something here?)