This question keeps cropping up, and whenever it does, people claim it was their language. The question needs to be precisely framed. Oldest spoken language in the world? All equally old. Oldest still spoken language in a particular area? It's easier to answer this for islands or archipelagos, e.g. Great Britain: Welsh. New Zealand: Maori. Japan: Ainu (but maybe not for much longer). Oldest written language? Sumerian…
What about languages we can only speculate about? E.g: the several languages of the Khoe-Sān people in South Africa. It is generally assumed that their language is, phonetically, the most "different" in the world and therefore we can suppose that it was one of the first to derive from the hypothetical "proto-language".
Interstingly, there are a number of words from these languages that are still widely used in South African slang: Dagga for Marijuana, Eina for "Ouch!", Gogga for Bug or Insect, and Kerrie for Stick, usually used as the neologism Knobkerrie. Very possibly some of the oldest slang in the world!