For example, the post about swords in medieval Europe had a comment of trade linking British Isles to India. How long ago could these connections have been established?
This question is probably best directed at /r/AskAnthropology. There lots of evidence in prehistoric sites of rare materials from far away: obsidian knives at great distance from the source of that obsidian, shells in inland sites, and so on. That's pretty good evidence of long-distance trade, or at least of hand-to-hand-to-hand trade long before civilization.
I don't know if it's the earliest evidence, per se, but lapis lazuli is found in in Naqada II contexts dating to ~3500-3200 BC. The nearest known source is in Afghanistan.
See: Cox 2012 "Trade and Power: The Role of Naqada as a Trading Centre in Predynastic Egypt" in Egyptology in Australia and New Zealand 2009: Proceedings of the Conference held in Melbourne, September 4th-6th available here.