In barter deals... Say, a farmer wants to exchange his chickens for French lessons. Do we say like " A farmer pays a tutor in chickens" OR... "A farmer pays a tutor with chickens". My guess is that BOTH forms are correct in spoken English at least. But I'm not a native speaker, so I'm not 100% sure about that.
"In" if the items used for payment are widely understood to be a token of exchange, similar to currency
The desk clerk presented me the bill which listed various currencies. I paid in US dollars.
They wanted 200 Swiss Francs. I paid in Euros.
The farmer didn't have cash to pay his neighbour to repair their shared fence, so he paid in chickens.
"With" if the items used for payment are not widely used as a token of exchange
The farmer tried to pay for his language lesson with chickens. "Quelle horreur!", exclaimed the startled lady, shooing them out of her house.
The expression "pay in chickens" is probably related to the term payment in kind which is defined by Longman's online dictionary as
a way of paying for something with goods or services instead of money
Using Google Ngrams to compare paid in chckens with paid with chickens indicates that, although both terms are very rare paid with chickens is rarer than paid in chickens. Repeating the Ngram analysis with "chickens" replaced by "fish", then with "wheat" and then other barter goods shows similar patterns, but with even fewer occurrences.
This suggests that, although both forms are rare, "paid in something" is rather more common than "paid with something", possibly because of the relation to "payment in kind".