Fix a field $k$. Does there exist a faithfully flat extension $A\subset B$ of $k$-algebras where $B$ is finitely presented but $A$ is not?
Edit: This question has been put on hold for missing context. I am not sure what context will help the question, but here are some other places where finite presentation can be detected within a faithfully flat extension:
If $A\subset B$ is a faithfully flat extension of rings, $M$ is an $A$-module, and $B\otimes_A M$ is finitely presented over $B$, then $M$ is finitely presented over $A$.
If $A\subset B$ is an inclusion of commutative $k$-Hopf algebras and $B$ is finitely presented as a $k$-algebra, then $A$ is finitely presented as a $k$-algebra.
These situations seem to be rather different from just an inclusion of $k$-algebras, and their proofs do not seem to shed light on this situation.
For all examples that I am familiar with of extensions $A\subset B$ of $k$-algebras for $B$ is finitely presented and $A$ is not, the extension fails to be flat (or I cannot prove it is so). For example, the inclusion $k[x,xy,xy^2,\ldots]\subset k[x,y]$ is discussed in the comments below, and the examples I am familiar with are of a similar form to this. (If we work with general commutative rings rather than $k$-algebras, I still do not know an answer to the question.)