This paper builds on top of Positselski (J Homot Relat Struct 19(4):635–678, 2024). We consider a complete, separated topological ring \({\mathfrak {R}}\) with a countable base of neighborhoods of zero consisting of open two-sided ideals. The main result is that the homotopy category of projective left \({\mathfrak {R}}\) -contramodules is equivalent to the derived category of the exact category of flat left \({\mathfrak {R}}\) -contramodules, and also to the homotopy category of flat cotorsion left \({\mathfrak {R}}\) -contramodules. In other words, a complex of flat \({\mathfrak {R}}\) -contramodules is contraacyclic (in the sense of Becker) if and only if it is an acyclic complex with flat \({\mathfrak {R}}\) -contramodules of cocycles, and if and only if it is coacyclic as a complex in the exact category of flat \({\mathfrak {R}}\) -contramodules. These are contramodule generalizations of theorems of Neeman and of Bazzoni, Cortés–Izurdiaga, and Estrada.
