Via Save As or Save a Copy, it doesn't seem possible to choose to save an artboard out of a range as an AI file. You can't even select "Use Artboards/range" when you try to save as AI. It's grayed out.
The only way I know right now to 'extract' an artboard is to save as EPS or some other format if I want to save only one artboard out of a range and retain my layers.
This seems a little odd that you can't save as AI directly..
Probably collecting everything on an artboard into a new layer might give you the option to use a script to save layers to files.
But this is a feature request but no as of yet has suggested a way that this might work, at least I do not think so.