Foreign support is the only option. Even so, it won't probably give rise to a Mapuche state, but to a French or Britich domitated area.
The Mapuche were able to adapt both some Incan tecnology (textiles, llamas and metalurgy -silbver-) and -later- some Spanish tecnologies (horse riding and Spanish warefare tactiques).
But, since they were mostly nomads, there's no way they could have been able to beat a modern state like 1860s Argentina or Chile. Nomad societies are by definition smaller than sedentary ones in terms of inhabitants. The Mapuche in Argentina relied for their survival on hunting, cattle, government food rations, trade and (mostly in Neuquen, but also elsewhere) a rudimentary agriculture. There's no way such a society could have produced gunpowder or modern guns. Even if they had been suplied by the French or the Brithish, numbers were against them (Chile and Argentina had much more population). And if the British or the French get involved directly, you'd likely get a British or French colony, not a free Mapuche state.
If serious problems (civil wars, etc.) had affected Argentina AND Chile severely in the 1860s and 1870s, and these had prevented the unificattion and the modernization of both nations that occured in those decades, the Mapuche might have survived a bit longer. But, if that had happened, they'd likely be subjugated by Europeans: their lands were very valuable and easy to occupy by any nation with access to modern 1860s technology. There weren't jungles or anything that could protect them (only the Andes and their cold forrests, but these are a very small part of Mapuche territory). There wasn't any comparable area that left "unclaimed" by 1880, and I don't see how it could have happened IOTL. Sombody would have claimed it if Argentina or Chilew hadn't.
Maybe with an earlier Pod? Possible, but bear in mind that it were the very same things that allowed the Mapuche to survived the Spanish conquest in the first place (lack of a centralized state, lack of cities, high movility, loose confederative organizations) which made extremly hard for them to survive independent during the second half of the XIX.
Still, if the Mapuche in Neuquen and South-Central Chile had depelopped a more advanced agriculture and gotten the full support of a European Great Power, they might have survived in that area. The Pampas are going to be lost anyway, and so is Patagonia. But maybe some sort of "French Belize" is formed in Southern Chile. Foir some reason, not many French move in. The Mapuche are still the mayority in the colony, but they have to learn French and other "French" stuff. Eventuall, by 1960, the colony is given independence, and a small Mapuche state comes into existance.