The only English loanword from Tocharian...

While the Oxford English Dictionary disagrees, the word shaman likely comes from Tocharian B. This is wild because we honestly don't know who the speakers of Tocharian B were. We know that the three Tocharian languages (labeled A, B, and C) were spoken in China. We also know they didn't survive past the Middle Ages. We also know that they are Indo-European, languages related to English, Hindi, and Greek - among others. We guess that it may have been the tongue of the Tarim people (a contemporary painting of them from 6th century is pictured). The Tarim mummies in China have red and blond hair and light eyes, but the link of the Tocharian languages to them is speculative.

Phonologically, the word shaman most closely resembles Tocharian B sama:ne (the first <s> being pronounced the same as the <sh> in English). The Tocharian word, in turn, comes from Sanskrit. The OED speculates that the Sanskrit word was distorted by an unknown Chinese language, which then entered into English.

Fortson, Benjamin W. Indo-European Language and Culture: An Introduction. 2nd Edition. Wiley-Blackwell. 2010.

"shaman" in Online Etymology Dictionary. 2014.