Well, almost everything has already being said. I don't know if Peru or Chile could have balcanized between 1700 and 1890. May be you can get a Mapuche state only in parts Southern Chile and Neuquen under forgein protection (that is much more possible to me than one that comprises all Araucania and Patagonia)
Argentina was in fact diveded between 1852 and 1859. The state of Buenos Aires had it's own constitution and its own foreign representatives. The rest of the Old 14 provinces formed a separate state (The Argentinian Confederation). If the autonomist party had being stronger in Buenos Aires during this period, and if Buenos Aired had won the Battle of Cepeda against the Confederation, this separation might have continue.
It's not certain however, because, as Juanmanuel said, one state may just have conquered the other. The Confederation had no sea port, and that was a source of inestability.
But, hypotetically, it is possible to have a scenario in which both states are divided, and compete for getting the lands in the pampas. Patagonia might then go to Buenos Aires (the most likely), to the British, to the French or to even the Welsh (if the settlement gets independent during a war between Buenos aires and the Confederation).