Beautiful game wonderfully executed.
Is it possible to save the game at all? (I could just be missing something)
Yep! In the bottom right you'll find a skip/next button. If the button says "skip" that means you haven't beaten the level. If it says "next" that means you have. Kind of a janky implementation, sorry!
There's a bug that makes it invisible during the tutorial, but you can still click on it.