Let $\kappa<\lambda$ be cardinals where $\lambda \geq \aleph_0$ and $\kappa>1$. Is there necessarily a set ${\cal S}\subseteq {\cal P}(\lambda)$ with the following properties?

1. ${|\cal S}| > 1$,
2. $|A| = \lambda$ for all $A\in {\cal S}$, and
3. $B\subseteq \lambda$ with $|B| = \kappa$ implies $\big|\{A \in {\cal S}: A \subseteq B\}\big| = 1$.

Let $\kappa<\lambda$ be cardinals where $\lambda \geq \aleph_0$ and $\kappa>1$. Is there necessarily a set ${\cal S}\subseteq {\cal P}(\lambda)$ with the following properties?

1. ${|\cal S}| > 1$,
2. $|A| = \lambda$ for all $A\in {\cal S}$, and
3. $B\subseteq \lambda$ with $|B| = \kappa$ implies $\big|\{A \in {\cal S}: B \subseteq A\}\big| = 1$.