Let $M$ be a saturated model of a theory $T$ in a first-order language $\mathcal{L}$, and let $N$ be a submodel of $M$.

Is it possible to have a substructure $A\neq N$ of $M$ such that $N \subset A \subset M$ and every element of $A$ is definable by a formula in $L$ with parameters from $N$?