  We introduce a dynamical Mordell-Lang-type conjecture for coherent sheaves.
When the sheaves are structure sheaves of closed subschemes, our conjecture
becomes a statement about unlikely intersections. We prove an analogue of this
conjecture for affinoid spaces, which we then use to prove our conjecture in
the case of surfaces. These results rely on a module-theoretic variant of
Strassman's theorem that we prove in the appendix.
