I booked a hotel, paid a deposit and arrived late. The hotel said the room was full and wanted to refund the deposit to us.

No, this deposit can be refunded to you. If you don't arrive at the specified time, the hotel has the right to regard you as giving up the right to stay in the hotel. In fact, from the hotel's point of view, it is easy to understand that the hotel relies on the room occupancy rate to maintain its operation. If it is empty, it will cost a day and the cost will increase accordingly. If you were the person in charge of the hotel, you would do the same. Please understand more.