Yes, you can have luxury staying in a cave. In the relatively unknown Italian province of Basilicato, there is a town called Matera . An ancient area of the town, known as the Sassi, is filled with intact trogladyte dwellings – homes, businesses, restaurants, hotels, etc. Here it is possible to sleep in a cave in luxury.
The area has been inhabited since paleolithic times and was recently declared a World Heritage Site. While it is challenging to drive into the old town and find parking, it is really worth the effort. It is amazing to walk along streets that are, in fact, the roofs of other dwellings.
There are lots of hotels in the Sassi but one, where I stayed, was simply amazing. The Hotel in Pietra (Hotel in Stone) offers lovely rooms set in caves. Sleeping in one is quite the experience.
If you are ever in this part of Italy, the Sassi is well worth a visit. You may recognize parts of it from films as it is often used to depict Jerusalum such as in Mel Gibson’s The Passion of the Christ (2004).