As far as I know, "mixin" is a Ruby concept for handling multiple inheritance limitations, and does not have a built-in comparable way of doing it in C#, VB.NET, or Java.
Your suggestion of adding extension methods to an interface is an interesting one, and would provide much of the benefit of a mixin. I've never actually seen anyone do it that way before, so I'll have to put that in my bag of tricks...
As far as an example, I think the one you put in your comment above isn't a bad one; demonstrating that a Bat is a mammal that can fly. Without making your own contrived example like that, I can't think of anyplace where this just "occurs" in any of the core .NET libraries, other than Linq (sort of, though even IEnumerable requires that you define the
GetEnumerator method in the implementing class, so it isn't a purely "empty" interface with only extension methods.)
Is there a problem with presenting your own contrived example?