Let $A$ and $B$ be noetherian normal rings and let $f:A\rightarrow B$ be a finite but non-flat ring homomorphism. We can also assume $Spec(A)$ connected if necessary. We put on $B$ the structure of $A$-module given $f$.

Is it true that $B\otimes_A B$ is normal?

The example I need to tackle comes from the following situation: take $A$ normal and noetherian, $K$ its function field, $K\subset L$ a finite extension, $B$ the normalization of $A$ in $L$.