The least ordinal $\kappa$ such that $L_\kappa(\mathbb R)$ satisfies KP is a counterexample. This essentially reduces to Theorem 1.3 of Tony Martin's "The Largest Countable This, That, and the Other." Of course, we are assuming some determinacy; $\text{AD}^{L(\mathbb R)}$ is more than enough, and Martin used the optimal hypothesis.

Here goes. Since KP is finitely axiomatizable, $L_\kappa(\mathbb R)$ is the least level of $L(\mathbb R)$ satisfying some sentence, as you have required. The sets that are (lightface) $\Sigma_1$-definable over $L_\kappa(\mathbb R)$ are precisely the inductive sets. There is a largest countable inductive set, which is just the set of reals $x$ such that for some $\alpha < \kappa$, $x$ is definable in $L_\alpha(\mathbb R)$. But by Martin's theorem, the set $C$ in fact contains every real definable in $L_\kappa(\mathbb R)$. (We are translating Martin's theorem from the notation of his paper, where the collection of reals definable in $L_\kappa(\mathbb R)$ is denoted by $\bigcup \Sigma^*_n$.) The complement $A$ of the largest countable inductive set is of course coinductive, or in other words $\Pi_1$-definable over $L_\kappa(\mathbb R)$. Yet $A$ contains no reals that are definable in $L_\kappa(\mathbb R)$: by Martin's theorem it is equal to the set of all reals that are not definable in $L_\kappa(\mathbb R)$. It follows that the definable elements of $L_\kappa(\mathbb R)$ do not form an elementary substructure of $L_\kappa(\mathbb R)$, since any elementary substructure of $L_\kappa(\mathbb R)$ would contain an element of the nonempty definable set $A$.

The least ordinal $\kappa$ such that $L_\kappa(\mathbb R)$ satisfies KP is a counterexample. This essentially reduces to Theorem 1.3 of Tony Martin's "The Largest Countable This, That, and the Other." Of course, we are assuming some determinacy; $\text{AD}^{L(\mathbb R)}$ is more than enough, and Martin used the optimal hypothesis.

Here goes. Since KP is finitely axiomatizable, $L_\kappa(\mathbb R)$ is the least level of $L(\mathbb R)$ satisfying some sentence, as you have required. The sets that are (lightface) $\Sigma_1$-definable over $L_\kappa(\mathbb R)$ are precisely the inductive sets. There is a largest countable inductive set, which is just the set of reals $x$ such that for some $\alpha < \kappa$, $x$ is definable in $L_\alpha(\mathbb R)$. But Martin's theorem states that this set in fact contains every real definable in $L_\kappa(\mathbb R)$. (We are translating Martin's theorem from the notation of his paper, where the collection of reals definable in $L_\kappa(\mathbb R)$ is denoted by $\bigcup \Sigma^*_n$.) The complement $A$ of the largest countable inductive set is of course coinductive, or in other words $\Pi_1$-definable over $L_\kappa(\mathbb R)$. Yet $A$ contains no reals that are definable in $L_\kappa(\mathbb R)$: by Martin's theorem it is equal to the set of all reals that are not definable in $L_\kappa(\mathbb R)$. It follows that the definable elements of $L_\kappa(\mathbb R)$ do not form an elementary substructure of $L_\kappa(\mathbb R)$, since any elementary substructure of $L_\kappa(\mathbb R)$ would contain an element of the nonempty definable set $A$.

The least ordinal $\kappa$ such that $L_\kappa(\mathbb R)$ satisfies KP is a counterexample. This essentially reduces to Theorem 1.3 of Tony Martin's "The Largest Countable This, That, and the Other." Of course, we are assuming some determinacy; $\text{AD}^{L(\mathbb R)}$ is more than enough, and Martin used the optimal hypothesis.

Here goes. Since KP is finitely axiomatizable, $L_\kappa(\mathbb R)$ is the least level of $L(\mathbb R)$ satisfying some sentence, as you have required. The sets that are (lightface) $\Sigma_1$-definable over $L_\kappa(\mathbb R)$ are precisely the inductive sets. There is a largest countable inductive set, which is just the set of reals $x$ such that for some $\alpha < \kappa$, $x$ is definable in $L_\alpha(\mathbb R)$. But by Martin's theorem, the set $C$ in fact contains every real definable over $L_\kappa(\mathbb R)$. (We are translating Martin's theorem from the notation of his paper, where the collection of sets definable over $L_\kappa(\mathbb R)$ is denoted by $\bigcup \Sigma^*_n$.) The complement $A$ of the largest countable inductive set is of course coinductive, or in other words $\Pi_1$-definable over $L_\kappa(\mathbb R)$. Yet $A$ contains no reals that are definable over $L_\kappa(\mathbb R)$: by Martin's theorem it is equal to the set of all reals that are not definable over $L_\kappa(\mathbb R)$. It follows that the definable elements of $L_\kappa(\mathbb R)$ do not form an elementary substructure of $L_\kappa(\mathbb R)$, since any elementary substructure of $L_\kappa(\mathbb R)$ would contain an element of the nonempty definable set $A$.

The least ordinal $\kappa$ such that $L_\kappa(\mathbb R)$ satisfies KP is a counterexample. This essentially reduces to Theorem 1.3 of Tony Martin's "The Largest Countable This, That, and the Other." Of course, we are assuming some determinacy; $\text{AD}^{L(\mathbb R)}$ is more than enough, and Martin used the optimal hypothesis.

Here goes. Since KP is finitely axiomatizable, $L_\kappa(\mathbb R)$ is the least level of $L(\mathbb R)$ satisfying some sentence, as you have required. The sets that are (lightface) $\Sigma_1$-definable over $L_\kappa(\mathbb R)$ are precisely the inductive sets. There is a largest countable inductive set, which is just the set of reals $x$ such that for some $\alpha < \kappa$, $x$ is definable in $L_\alpha(\mathbb R)$. But by Martin's theorem, the set $C$ in fact contains every real definable in $L_\kappa(\mathbb R)$. (We are translating Martin's theorem from the notation of his paper, where the collection of reals definable in $L_\kappa(\mathbb R)$ is denoted by $\bigcup \Sigma^*_n$.) The complement $A$ of the largest countable inductive set is of course coinductive, or in other words $\Pi_1$-definable over $L_\kappa(\mathbb R)$. Yet $A$ contains no reals that are definable in $L_\kappa(\mathbb R)$: by Martin's theorem it is equal to the set of all reals that are not definable in $L_\kappa(\mathbb R)$. It follows that the definable elements of $L_\kappa(\mathbb R)$ do not form an elementary substructure of $L_\kappa(\mathbb R)$, since any elementary substructure of $L_\kappa(\mathbb R)$ would contain an element of the nonempty definable set $A$.