Let $K$ be an extension field of a field $k$. We say $K$ is separably generated over $k$ if $K$ has a transcendence basis $S$ over $k$ such that $K$ is separably algebraic over $k(S)$.
Let $k$ be a perfect field of characteristic $p \neq 0$. Let $K$ be an extension field of $k$. It is well-known that if $K$ is finitely generated over $k$, $K$ is separably generated over $k$. I would like to know examples of $K$ which is not separably generated over $k$ in the following cases if any.
(1) tr.dim $K/k < \infty$
(2) tr.dim $K/k = \infty$