It might also be helpful to brainstorm a few characteristics of good managers, such as communication, leadership, teamwork, strategic thinking, etc.
Once you've got your list, think of experiences where you've used these skills. I think if you have something very compelling that isn't professional, go for it. But keep in mind that you need to make the connection very clear - many people deal with strife of some kind, but won't make good managers. Show them why you will, and why the situations supports your claim.
I would agree that professional situations would seem to provide the clearest links, but if you can make your case in another situation, go for it.