For any cardinal $\kappa$ at least the size of the continuum, the "really long line" of length $\kappa$ is an example.

This space, let's call it $L_\kappa$, is defined as follows. Begin with the ordinal $\kappa$ with its usual order topology. Then, for any $\alpha \in \kappa$, connect $\alpha$ and $\alpha+1$ with a copy of the unit interval.

$L_\kappa$ obviously has cardinality $\kappa$, and a discrete subspace of size $\kappa$ is given by the set of all successor ordinals.

If $\kappa < \mathfrak{c}$, then you can get an example by modifying the construction of $L_\kappa$: simply replace the unit interval with a countable connected Hausdorff space. (It's not obvious, but countable connected Hausdorff spaces do exist.)