an interesting article I recently came across, about the introduced population of nilgai found in the wilds of Texas. Nilgai: Blue Antelope of the Anthropocene | Cool Green Science
Yes someone else showed me the article. I was in the area (including Laguna Atascosa) last November and sort of hoped to see one, but did not. (I even more wanted to see an ocelot, but no luck there either).
Nilgai and Blackbuck both exist in free-roaming populations. In New Mexico, the Gemsbok, has become established. Several other species of antelope (most notably Scimitar-horned Oryx and Addax) can be found confined to large ranches. Aoudad, Wild Boar (or feral hog), Sika Deer, and Axis Deer are fairly common as well. Don't know if they are breeding, but I've seen Emu in some strange places too.