A corollary of Schur’s lemma in positive characteristic

Question

A corollary of Schur's lemma reads (nLab):

In the case that the ground field is an algebraically closed field of
characteristic zero; endomorphisms $\psi:V \rightarrow V$ of a finite dimensional
irreducible representations V are a multiple $c \cdot \textrm{Id}$ of the identity
operator.

The proof given in section 3 of the nLab page does not explicitly state that it uses the characteristic being zero.

1. Where does the proof fail when the ground field is algebraically closed but of positive characteristic?
2. What are some positive characteristic counter-examples?
3. Is there a weaker form that works in positive characteristic?

Answer 1

It works perfectly well in arbitrary characteristic, with the same proof : over an algebraically closed field, in finite dimension, $\psi$ has an eigenvalue $c$ thus $\ker(\psi-c.id)\neq 0$ is a subrepresentation, therefore it is the whole of $V$, so $\psi = c.id$