On the following page of our guide to Minecraft: Dungeons you can find more information about saves and how the saving system works in the game.
Can I save the game while I'm playing on a map?
No. You have to complete every map from the beginning to the end. You can only save the game when a particular map has been completed.
How to save the game?
When you finish a map in Minecraft: Dungeons, you can just quit the game. The game saves your progress on a regular basis. If you finish the map and just leave, all your progress will be saved.
Remember that you can repeat and replay all maps. That way you don't have to be afraid that you might just have overwritten a fight or a map. You can go back to another location at any time.