Per the Mirror's Paul Brown, a man threw the skin at the striker after he opened the scoring with an early penalty.
Police are investigating the incident to determine if it was racially motivated, while the Football Association is also looking into the matter.
Per Sky Sports, the supporter has been issued an indefinite ban by Tottenham after a club spokesperson said following the incident: "Behaviour such as this is completely unacceptable and the supporter in question will be issued with a ban."
BBC Sport's Simon Stone provided further insight into the FA's investigation:
Simon Stone @sistoney67
Expect Arsenal + Tottenham to be charged for failing to control players in melee that followed Eric Dier's goal. Action against Dier less certain. Person who threw banana skin is a police matter. FA speaking to club and Kick it Out. Will endorse any action against the fan.#ARSTOT
Including the fan who threw the banana skin, Metropolitan Police arrested seven people for public order offences during the derby on Sunday.
Two of the arrests are thought to be Arsenal supporters who lit smoke canisters inside the stadium.
Tempers flared on the pitch during a tempestuous encounter between the bitter rivals.
Eric Dier sparked a heated confrontation between both sets of players after he motioned to silence the crowd when he made the score 1-1 in the first half, during which objects were thrown from the stands.
Jan Vertonghen was also sent off late on when he received a second yellow card for a foul on Alexandre Lacazette.