The anthropologist David Graeber argues in Debt: The First 5000 Years that trade starts with some sort of credit namely the promise to pay later for already handed over goods. Therefore, credit and debt existed even before coins.
The word "debt" comes from the French dette and ultimately Latin debere (to owe), from de habere (to have). The letter b in the word debt was reintroduced in the 18th century, possibly by Samuel Johnson in his A Dictionary of the English Language (1755), as several other words that had existed without a b had them reinserted at around that time.