Given a finite dimensional connected algebra $A$ with $Ext^{1}(M,M) \neq 0$ for any non-projective and non-injective indecomposable module $M$, with the condition that at least one such module exists.

Is $A$ selfinjective?

Is $A$ local?

(answer is no,see the answer by Jeremy Rickard)

Two other questions:

Is it local when it is selfinjective?

Is it selfinjective when it is local?