The incident occurred off the Russian-occupied Crimean Peninsula. According to the Coast Guard, the attack took place off the coast of Sevastopol.
A fishing boat on site evacuated the five fishermen from the sinking ship and took them to the port of Inebolu on Turkey's Black Sea coast.
"Unfortunately, one of the injured fishermen, whose condition was critical, died during the evacuation," the Coast Guard wrote in a statement on Friday.
Turkey warned at the end of May of an "uncontrollable escalation" of attacks after a Turkish cargo ship was attacked by drones.
The Ukrainian navy said a Russian drone carried out the attack.
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