If $(X,\tau)$ is a topological space, we say $S\subseteq X$ is discrete, if the subspace topology on $S$ inherited from $(X,\tau)$ is discrete.

Is there a connected $T_2$-space $(X,\tau)$ and a discrete subset $S\subseteq X$ such that no proper superset of $S$ is discrete?

If $(X,\tau)$ is a topological space, we say $S\subseteq X$ is discrete, if the subspace topology on $S$ inherited from $(X,\tau)$ is discrete.

Is there an infinite connected $T_2$-space $(X,\tau)$ and a discrete subset $S\subseteq X$ such that no proper superset of $S$ is discrete?

EDIT: Added "infinite" in the question.