I'm sorry. This is a bug. There is a save and even an auto-save, but you need to go to the menu to save (press Escape or the power button on the top left to get there). The game will automatically save to your selected slot when you go to the menu. You can select another slot and press Save to overwrite it as well.
I will fix this in the next build. For now, please go to the menu before closing the browser.