The big three ruminants that are typically used for dairy are well known: cows, ewes, and goats. Some people are aware that mozzarella and some other cheese are typically made from buffalo milk as well. But there are several other animals that are used for dairy in some parts of the world, if not for direct drinking of milk, they are used for cheese and yogurt.
There is also some use of hogs for milk, although the reportedly strong flavor turns most people off.
The logistics of using some of these animals for milk and cheese cannot be covered here, and it should be noted that it is likely difficult to get high enough production to make anything more than smaller and fresher cheeses. Ewe (sheep) milk is known to freeze well, and so can be collected over multiple milking sessions, and it is possible that some of the above more rare dairy animals' milk can similarly be frozen without much loss in quality so it can be used in more serious cheese making endeavors.