Is torsion of a topological module closed?

Question

I was asking to myself the following question. Consider a $p$-adically complete and separated topological algebra $R$ over $\mathbb{Z}_{p}$. As $\mathbb{Z}_{p}$ is a domain, we know that the $\mathbb{Z}_{p}$-torsion elements of $R$ form a submodule. Is it possible to prove that this torsion is a closed submodule of $R$? I'm interested in the possibility of saying that the quotient of $R$ over its torsion is again $p$-adically complete, but in order to do this, I think I need that the torsion submodule is closed. Have you got any idea? I tried to prove it by hands but I didn't reach anything useful. Thank you very much!