Indeed, I suspect it would. The (SPX3819 Low Drop Out) regulator is great for it's ability to provide full output at 3.6v, and does have a maximum input voltage of 16v, but Erik noted that with load presented by the Oak, it was only good to around 7v before it gets too hot, and is likely to reset due to thermal protection. So there is a good chance that yes, it will run from a 9v battery, but how long it runs before it will starts resetting and causing the Oak to do strange things I don't know. If you can heatsink it, it will probably be ok, but the reality of the situation is you're not going to get much run-time from a 9v battery anyway - much better to consider running it from a 5v USB powerbank if you want that sort of portability and ease of use.