Others have indicated that this is not possible. I'll defer to them on the specifics of the language. This answer is more "what to do since it doesn't work."
Obviously, your options depend on your application. I've seen enums most often to give a name and naming scope to an otherwise arbitrary integer value (as opposed to using macros). If this is your usage, you could take the alternative of building a dictionary (or pair of dictionaries) to maintain indexes to these names and values. In your base class, you'd populate it with the default values. In child classes, you add to it the same way. In grandchild classes, you add to it the same way. in great-grandchild classes, you add....
The overhead of such a system should be minimal, since you are just hashing relatively short strings as names, in addition to store integers. You do take a hit at run-time compared to using a compiled-in comparison of int values, but the lookup should be negligible unless you are on a real-time (unlikely if C# is your language) system or an enterprise system.