Yes, the Lakota and the Apache did encounter each other, but direct battles between the two groups were relatively rare. The Lakota, part of the Great Plains tribes, and the Apache, native to the Southwest regions, had different territories and lifestyles, which often kept them apart.
However, there were instances where their paths crossed, especially as both groups faced pressures from European settlers and U.S. military forces encroaching on their lands. Competition for resources, such as hunting grounds and always scarce water supplies, occasionally led to conflict. The dynamic nature of tribal relations during this period means that while they were not traditionally enemies, skirmishes could occur when their interests clashed.
The historical context of each tribe is also important. The Lakota were more focused on buffalo hunting on the plains, while the Apache were often engaged in raiding and defending their territories in the rugged Southwest. This difference in lifestyle greatly influenced the nature of any potential encounters. Overall, while there might have been some moments of conflict, it wasn’t a continuous engagement like those seen within their own respective conflicts or with other tribes.