I don't think there are any ring languages, and I can explain why it's unlikely.
Remember that language evolution is only roughly the same as biological evolution, and the line between languages is only roughly the some as the line between species.
One big difference is that there just aren't nearly as many living languages as there are extant biological species, by orders of magnitude. Since the classic example of ring species is the Larus herring gulls, let's look at them as an example. There are many Arctic bird species. A few of them cover the whole circle. Of those, one, the Larus gulls, form at ring species. By contrast, there are only a handful of Arctic languages, and the widest-distributed small family is Eskimo-Aleut, which doesn't go any farther than from the Russian Aleutian Islands to eastern Greenland. So there's no opportunity for a ring language to form.
Also, many species share the same environment, in different niches, but languages are mostly all competing for the same niche. The fact that herrings spread around the Arctic means it's more likely for herring gulls to spread into the same territory. By contrast, fact that Samoyedic languages spread around the Arctic means it's less likely for Tungusic languages to spread into the same area.
So, the most widespread language family around the Arctic ends up being Eskimo-Aleut, which only goes from the Russian Aleutian Islands to eastern Greenland. (In Siberia, where there's more competition, the continuua are much narrower than that.)
There are places where you can find smaller rings of species, like your Himalayan example, but the same factors, and others, make it less likely to find rings of dialects.
Let's look at an example that superficially looks promising: Scandinavia. Finnmark Nordnorsk and Far Norrlander Swedish dialects are not mutually intelligible. But there is a chain of pairwise-intelligible Norwegian dialects all the way down and around the coast of Norway, and a chain of mutually-intelligible Swedish dialects all the way down the coast of Sweden. At the bottom, the southern varieties of Standard Norwegian are pretty close in intelligibility to Scanian Swedish, to the point where (if you wanted to annoy a lot of people) you could almost claim they were dialects of Danish.
But as soon as you look beyond the coasts, you'll see that the traditional inland northern Swedish and Norwegian dialects share features with each other. When species send out branches into thinly-populated areas, they tend to lose contact and diverge, but when languages do, they're carried by people who move back and forth.
Also, there's a constant pull on both dialects by their respective standard varieties—especially strong in recent times—which is part of the reason they're as divergent as they are.
Also, some of the Finnmark dialects aren't divergent just because Nordnorsk speakers keep moving farther north; much of the change comes from other people (mostly Sami) learning Norwegian as a second language, a kind of "horizontal transmission" that you don't see in higher animals in biology.
So, a ring seems likely to form only if the center of the ring really is nearly uninhabitable, if the language doesn't have writing and mass media, and if there's nobody else in the way.
Such a situation could arise somewhere in, say, the highland Trans New Guinean languages. But it would happen a lot less often than with biological species. And, even if it did, there's a good chance we'd never find out—the only way we could detect a highland TNG ring would be if we discovered and classified all of the languages before contact changed the map.