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 $N$?



Saturated models and definable substructures
Yes. Let $T$ be the theory of an endless discrete order, which is a complete theory. Let $M=\mathbb{Z}\cdot\mathbb{Q}$ consist of $\mathbb{Q}$ copies of the $\mathbb{Z}$ order, which is a countable saturated model of $T$, and let $N$ consist of only the even elements in each copy of $\mathbb{Z}$, which still forms a model of $T$ (though not an elementary substructure of $M$), with $A$ adding in some or all of the odd elements. Each of these new elements in $A$ is definable in $M$ from their immediate predecessors, which are in $N$. 


