If I exit the game during a battle, can I resume it later?

From my experience:

I was in a battle, then I closed it. When I re-opened the page, the page said this:

When I clicked "Yes", I returned to exactly where I was.

Edit: This answer is outdated. It is now possible to resume a battle (See the accepted answer).

Exiting the game before finishing a battle will count as a failure. This means that you'll need to replay the battle, provided you have some attempts left. If the battle was part of a larger adventure module (eg. White Skull Canyon in the example below), your progress in the module will be preserved. You will just have to replay the battle that you were doing when you closed the browser. Here's the screen that I saw when I closed my browser during a battle and then started the game again: