Harmon "Buck" Bokai was a Human baseball player during the 21st century, considered by many to be one of the greats. Bokai was born on October 31, 1998 in Marina del Rey, California. (DS9: "The Storyteller")
Playing for the London Kings, Bokai began his career in the Major Leagues at shortstop, his rookie year being 2015. Some considered him the only asset of the Kings that year. (DS9: "Past Tense, Part II")
In his first three years in the Major Leagues, Bokai, a switch hitter, hit twenty home runs a year right-handed. In his fourth year, Old Crow moved Bokai to second in the batting order and he never hit more than ten home runs a year batting right-handed. (DS9: "If Wishes Were Horses")
In 2019, Bokai began his rookie year in the Planetary Baseball League for the Crenshaw Monarchs. The next year, he was the pivotal figure in a twelve-man trade with the Gotham City Bats, where he played two years. He played a season for Tanis and then two seasons with Seibu before signing with the London Kings in 2025. That year, he was two games shy of beating Joe DiMaggio's consecutive game hitting streak. In early 2026, Bokai was #311 in the "Greatest Heroes of the Planetary Baseball League" baseball cardset, printed by the Planetary Baseball League, Inc. (DS9: "The Storyteller")
Later in 2026, while playing shortstop for the Kings, Bokai broke the record for most consecutive games with a hit in a single season, which had previously been set by Joe DiMaggio in 1941. (TNG: "The Big Goodbye") He broke the record with a "squeaker" that went just under Eddie Newsom's glove. (DS9: "If Wishes Were Horses")
In 2032, while a third baseman, Bokai broke another of Joe DiMaggio's long-standing records in Game Five of the World Series. Despite this, the New York Yankees won the series in game six. (VOY: "One Small Step")
Bokai remained with the Kings throughout his career, until 2042, when the team made it to the final World Series. With the decline of the sport, only 300 spectators attended the last game of the series, which was won through a home run, hit by Bokai. Even though he thought he could have lasted five more years, Bokai retired from baseball soon after the series. (DS9: "If Wishes Were Horses")
An alien who was on a mission of exploration and had followed a ship through the Bajoran wormhole to Deep Space 9 appeared to Jake and Benjamin Sisko in the form of Buck Bokai in 2369. His people wanted to learn more about humanoids and became interested in the concept of imagination. (DS9: "If Wishes Were Horses")
According to the Star Trek Encyclopedia (4th ed., vol. 1, p. 90),
"The player who broke DiMaggio's record was mentioned in "The Big Goodbye" (TNG), but he remained nameless until "If Wishes Were Horses" (DS9). "The Big Goodbye" establishes that Bokai broke DiMaggio's record in 2026, but "One Small Step" puts the event on October 19, 2032, during game five of the World Series. We suspect that the graviton ellipse probably has something to do with the inconsistency. The character's name actually originated in a baseball card proposed by Star Trek: Deep Space Nine illustrator (and baseball fan) Ricardo Delgado as a decorative item for Ben Sisko's desk. Fellow baseball fan (and executive producer) Michael Piller suggested the card feature a 21st-century player, which would make it a valuable collectors' item to the 24th-century Ben Sisko. Star Trek model maker Greg Jein (yet another baseball fan) got into the act at this point, providing photos of himself in a baseball jersey that were converted into a prop card. Greg also provided the fictional "history" of his character and the statistics that appeared on the card. Bokai, whose name was a vague allusion to Buckaroo Banzai, from the movie of the same name, was mentioned in "The Storyteller" (DS9), but not actually seen until "If Wishes Were Horses", in which Bokai was played by actor Keone Young, who bore an uncanny resemblance to Jein. The prop baseball card, which was revised after the episode, has Young's photo on the front, but still shows Jein on the back."
Much of this information was repeated in the Star Trek: Deep Space Nine Companion (p. 54)
The front and back sides of the prop baseball card was seen in the Star Trek Encyclopedia (1st ed., p. 31). The Jein card and Sisko's Bokai baseball were later added as a display items to "The History of the Future Museum" exhibition part of the Las Vegas Star Trek: The Experience attraction. 
In "If Wishes Were Horses", Jadzia Dax stated that Buck Bokai died "two hundred years ago", which would place his death around 2169. That seemed unlikely, as he would have been 171 years old. Later in the episode, the alien impersonating Bokai stated he died two hundred years before Sisko was born, which was around 2132. He would have been 134 years old then.