Are there human-ingestible liquids that do not contain water?
Is there anything a human can drink that does not contain water? 


Yes, there are a lot of liquids that do not contain water, but mostly they are not safe for human consumption.

Mercury is a liquid at room temperature. Can a person swallow it? Most assuredly. Motor oil has no water in it. It's not a satisfying beverage. Canola oil, olive oil, and many other food derived oils can be imbibed in even reasonably large amounts, but it won't do your digestive tract a big favor. 

I've treated goats with propylene glycol (a liquid), parents treat their kids with mineral oil for constipation, etc. etc.

It depends on what you mean by "drink".