The title pretty much says it all.
I would like to know if it is true that given a finite flat ring extension $A \rightarrow B$ with $A$ a DVR, and $B$ a local ring, then $B$ is necessarily a domain. If it's not true, are there additional hypotesis on $B$ such that this is true?
Just to get this off the unanswered list.
No, it is not true that if $B$ is a flat local $A$ algebra then $B$ is a domain. The example of $B=A[x]/(x^2)$ shows this. That said, if you demand that $B/A$ is unramified then this is true. Indeed, since $A\to B$ is both flat and unramified it's etale. This implies that since $A$ is regular local that so then is $B$ (e.g. see Tag025N). But, regular local rings are domains (e.g. see Tag00NP).