Example of $\kappa$-closed but not $\kappa$-directed closed poset

Question

A poset $(\mathbb{P}, <)$ is $\kappa$-closed iff every descending sequence of length $<\kappa$ has a lower bound. $(\mathbb{P},<)$ is $\kappa$-directed closed iff every directed subset of size $<\kappa$ has a lower bound. If $\mathbb{P}$ is $\kappa$-directed closed, then it is $\kappa$-closed. I remember reading that the converse does not hold. What would be an example to show that?