Yes, it is possible! We stayed last year in june on the Elba Island, in Italy , Tuscany, for a week in a little village, called Sant'Andrea, and there was a little hotel (only 20 rooms) , with absolutely first class cuisine. They did only half board, but it was better, because we went out to lunch in other restaurants on the island.
If you will go there, I suggest you to book for 5 or 6 nights, and enjoy the excellent dinners. You can choose between sea fruits, fish or meal or vegetarian food , amazing wine and regional specialties . The dishes are served with style and the ingredients are fresh and local.
Look at their web site with pictures and then book a vacancy… www.hotelilio.com