Is there example for $f\colon A\to B$ being ring map, but the image $f^*\colon \operatorname{Spec}(B)\to \operatorname{Spec}(A)$ not constructible? (i.e., written as a finite union of locally closed subsets)
(Since Chevalley's theorem asserts it is true for $B$ finitely presented over $A$, an counterexample must be in the case $B$ is not finitely presented over $A$.)