Yes, that's right, a dairy.
I know that in most English-speaking places a dairy is a place that makes and supplies milk products, but in New Zealand it's a shop that's open outside usual trading hours.
It's a lovely use of the term, if likely to sow confusion among non-natives.
If you live in a house too small to have its own dairy in which to churn your cream into butter - which is basically, let's face it, all of us - then anything made of milk is termed dairy. This use is usually encountered nowadays as I don't eat dairy, a deeply irritating if efficient phrase managing to signal smugness, superiority, and entitlement in a very few words.
(Yes, there are poor people who really can't, but in this (and every) case a brief but sincere apology for causing trouble is due, and will soothe many a savage breast.)
Anyway, milk, butter, cheese, ice cream, cream, are all dairy products, and the animals which give us the milk to produce them are dairy animals.
There are dairy cattle:
Nigerian Dwarf Dairy Goat, photo by Jmkarohl
By Schaeferhof - Self-published work by Schaeferhof, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=1508043
By Jjron - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=2831408
and even donkeys:
Photo by Aloisio
But best of all is the derivation of the word itself, which is, so pleasingly, absolutely nothing to do with dairy products at all.
Word To Use Today: dairy. This word comes from the Old English dǣge, a servant girl, that is, one who kneads bread (it's basically the same word as dough).