Wool does not melt or drip
This answer might surprise you: wool!
Wool (...) does not melt or drip(.)
Wool ignites at a higher temperature than cotton and some synthetic
fibers. It has a lower rate of flame spread, a lower rate of heat
release, a lower heat of combustion, and does not melt or drip; it
forms a char which is insulating and self-extinguishing, and it
contributes less to toxic gases and smoke than other flooring products
when used in carpets.
From section "characteristics"
It's even quite commonly used a so called "fire-retardant material" in many end consumer textile products. Check out a list here.
Here's another interesting article regarding this topic:
According to Wool Gatherer, one thing that makes carded wool flame
resistant is the fact that the fibers are so close together that they
create an environment that is fairly devoid of oxygen thus resisting
flame. An example of this idea, although one I have not personally
tried and would not recommend, would be attempting to light a
phonebook on fire. Apparently, it is not easy to light a phonebook on
fire, as the pages are so tightly held together there is not enough
oxygen present to support the flame.
Last but not least you may want to watch a comparison between synthetic products and wool here: https://www.youtube.com/watch?v=WKuAl_HzCjs
As a product recommendation I can refer to this one (however, it has some cotton on the inside, the shell/outside is completely made out of wool):
This coat originated in the 1800s and is still as popular today. The
rugged wool-blend shell wards off the fiercest weather. It's not only
warm, but it sheds water, and the tight weave keeps out bone-chilling
wind. Two-piece sleeve for a roomier fit; knit storm cuffs in sleeves.
Four outer snap pockets, plus hand-warmer pockets and internal
security pocket. Back stowage pocket originally designed to put game
in. Under-collar tab keeps collar up to hold out cold and breezes.
Fully lined body and sleeves. Button front covered by storm flap.
You might have already seen this question. Please check out ShemSeger's answer.
Campfire clothes are the heavy wool jackets, and flannel shirts worn
by mountain men, hunters and lumberjacks. I have a wool jacket from
nepal that I use as my sit around the fire jacket.
First-aid responders will tell you that synthetic materials are
actually one of the worst things to have on when working with a fire,
because they will melt to your skin if they do catch.
This list from Wikipedia (already linked above) contains every material which come into question:
Fire-retardant materials used in textiles
Nomex (a DuPont trademark)
Indura FR Cotton (a WESTEX Tradmark)
Arselon (Khimvolokno trademark)
CXP(a WESTEX Tradmark)
Ultrasoft FR Cotton(a WESTEX Tradmark)
Pyrovatex fr cotton
TrueComfort(a WESTEX Tradmark)
Ultrasoft AC FR Cotton (a WESTEX Tradmark)