Looks a lot like this one:
http://www.amiga.org/gallery/index.php?n=3323You can replace the battery. Then your Amiga will remember the time & date.
After you've removed the old battery, you must neutralise the corrosion with lemon juice (do that whole half of board or whole board) and then rinse with water (you can put it under the tap). I'd used cotton wool buds to apply the lemon juice. You can also use concentrated lemon juice if you like.
Edit: Due to your chips being socket, might not be a great idea to wet those.
But in any case, do make sure all is nice and dry before turning back on again.