Let A be a valuation ring. Let A→B be a ring map of finite type. Let M be a finite B-module.
- If B is flat over A, then B is a finitely presented A-algebra.
- If M is flat as an A-module, then M is finitely presented as a B-module.
See Tag 053E.
PS: Much more is true, see the this chapter in the stacks project. The proof of the lemma above however is quite easy.