Fleshing out Will Brian's suggestion (and saving him the trouble of writing it up):
If $\kappa$ is finite you topologize $\lambda$ using the base $[\kappa-1,\lambda)$ (which is a perverted way of listing all the initial of $\lambda$ that contain $\kappa-1$. The initial segment $\kappa-1$ ensures that there is a set of size $\kappa-1$ that is not split; it and the other initial segments help to split all sets split all sets of size $\kappa$ or more.
If $\kappa=\omega$ you have to be a bit more careful: take $\{2^n:n\in\omega\}\cup[\omega,\lambda)$. The $2^n$ are needed to split all infinite subsets of $\omega$ and they are spread out enough to ensure unsplit subsets of arbitrary large finite cardinality. The rest ensures every infinite set is split.
If $\kappa$ is an infinite successor cardinal, say $\kappa=\mu^+$ then Will Brian's base works: $\{\mu\}$ together with all singletons above $\mu$. Continuing the perverse streak: $[\mu,\lambda)$ works too. In either case the set $\mu$ (and its subsets) is unsplit, everything with points above $\mu$ is split.
If $\kappa$ is a limit cardinal then one can let $A$ be the set of cardinals below $\kappa$ and use $A\cup[\kappa,\lambda)$ as a base. For every cardinal $\mu<\kappa$ the interval $[\mu,\mu^+)$ is unsplit; all subsets of $\kappa$ of cardinality $\kappa$ are split by the mebmbers of $A$.the rest ensure splitting of anything with points above $\kappa$.
