Is there a notion of *$p$-adic Dedekind Domains*?

Question

As we all know, the ring $\mathbb{Z}_p$ can be constructed as the projective limit of the rings $\mathbb{Z}/p^{n}\mathbb{Z}$.
Now is there any generalization such as the $p$-adic completions of a Dedekind Domain?
This might be said to be inspired by the general treatment of extensions of Dedekind Domains from the treatment of algebraic number fields.
In any case, thanks very much.