Arrow season 7 episode 5, "The Demon," has all but confirmed that Batman exists in some form in the Arrowverse. While The CW's ability to use Batman and his associated characters has been limited, there have been more than enough vague hints in the past to suggest that Batman exists on the same Earth as Arrow, The Flash, and Legends of Tomorrow. This new episode, however, has offered the most direct evidence yet.
The past few episodes of Arrow have seen the incarcerated Oliver Queen searching for the Demon - a mysterious inmate held on the lowest level of Slabside Penitentiary, who Oliver believes has information that can lead him to the crime-boss that is threatening his family. The Demon finally reveals herself to Oliver as Talia Al Ghul - his mentor and former ally. When Oliver asks what she is doing in prison, she sighs and says only that she "ran afoul of an old foe in Gotham."
Fans of the Batman comics are likely familiar with the character of Talia Al Ghul, who was originally introduced as a romantic interest for Bruce Wayne, who was torn between her love for the Dark Knight and her loyalty to her father, eco-terrorist Ra's Al Ghul. In time, she would come to reject both men, taking over a League of Assassins splinter group, called Leviathan, to destroy both her father and her lover. The character's background in Arrow is slightly different, with Talia having abandoned the League of Assassins to act as a teacher to those who sought to fight for a better world. It was in this capacity that she trained Oliver Queen and gave him the idea of creating a second persona to "become someone else ."
The suggestion that Talia was jailed after a fight with Batman is not the first time the show has hinted at a vigilante in Gotham City or the existence of Bruce Wayne. In the season 6 episode "Tribute," while dismissing a faked photo that seemed to show him dressed as Green Arrow, Oliver Queen joked about the possibility of Bruce Wayne being a masked hero. The Flash has also made reference to Bruce Wayne existing, with news of a merger between Queen Inc. and Wayne-Tech appearing below the fold on a newspaper front page from the future.
It's worth noting that Batman has been more firmly established to exist on Earth-38 - the parallel Earth on which Supergirl is set. James Olsen told Winn Schott that his worst fear as the vigilante Guardian was that people would come to see him "like Clark's friend" - a statement that caused Winn to knowingly point his fingers up on both sides of his head in imitation of Batman's cowl. Supergirl herself has made reference to Batman on two separate occasions, talking about a vigilante her cousin worked with occasionally who had "...tons of gadgets..." and "...lots of demons."
It's also worth mentioning that this is almost certainly a reference to Batman and not Batwoman, who is scheduled to make her Arrowverse debut in a few short weeks as part of the Elseworlds crossover event. Given Talia's long history with Batman as an enemy and a romantic partner, there would be little point in the shout-out to "an old foe in Gotham" if she had been referring to someone else. Sadly, while this may confirm that Batman does exist on Earth 1, there's next to no chance of him appearing in the flesh anytime soon. CW President Mark Pedowitz has gone on record as saying the network has no plans to introduce Batman on Arrow or even in the potential Batwoman series. While this news is sure to disappoint Batman fans, it's still heartening to know that the Dark Knight is out there.