Theorem 28.15 in Warner's "Topological Rings" book states a well-known characterization of the linearly compact modules over a topological ring $A$. More precisely:
A Hausdorff linearly topologized $A$-module $E$ is linearly compact if and only if $E$ is complete and all the quotients $E/U_\lambda$ modulo the the open submodules $U_\lambda$ in the fundamental system of neighborhoods of zero are linearly compact when equipped with the discrete topology.
Neither the book nor the classical literature I've seen so far seem to provide a counterexample for a linearly topologized complete module $E$ which is not linearly compact. In principle, I would need for a fundamental system of neighborhoods of zero $(U_\lambda)_\Lambda$ such that $E/U_\lambda$ is not linearly compact for some $\lambda$. However, I was not able to find it. I guess one example might be very easy to exhibit, but I'm not an expert in topological groups and rings, so some hint would be very appreciated.
Thank you!