NAnt's copy task lets you copy files from one location to another (which should do the majority of what you need for an ASP deployment?).
I've used it to push out ASP sites (e.g. grab latest from source, copy to drop location) as part of a larger build cycle that also was building ASP.NET 1.1. Granted, that was on v0.85 a few years ago, but that's a production example of deploying a mixed bag at the same time.
If you really have to do something that isn't part of the built-in tasks or you can't find one in the community, there's nothing preventing you from writing your own task. If you can do it in the .NET framework, you can probably do it as part of a NAnt build.
So, short answer: Yes, you can, but if you have anything more complicated than a basic build, you might have to write a custom task.