"The Ancient West," known during the 20th and 21st centuries as "the Wild West" and "the Old West," describes the period of 19th century Earth when the western United States was being settled. This period in Earth's history would be frequently romanticized in Earth literature for centuries.
In 2153, the Enterprise NX-01 discovered a Human and Skagaran settlement on a planet in the Delphic Expanse that resembled the towns of Earth's Ancient West. (ENT: "North Star")
Kirk, Spock, McCoy, Scott, and Chekov in an incomplete recreation of 19th century Tombstone, Arizona.
In 2268, the Melkot trapped James T. Kirk and his landing party in an incomplete recreation of the American Ancient West town of Tombstone, Arizona, where Kirk and crew were to be executed in a reenactment of the infamous gunfight at the O.K. Corral. (TOS: "Spectre of the Gun")
Deanna Troi was a fan of stories set during the Ancient West. Her father, Ian Andrew Troi, often read her such tales when she was a child.
Deadwood, South Dakota, 19th Century Earth
In 2369, Worf, his son, Alexander Rozhenko, and Deanna Troi participated in a holodeck adventure set during the Ancient West in the town of Deadwood, South Dakota. The program was designed by Alexander and Reginald Barclay. Troi, Worf and Alexander played this program when an experiment being conducted by Geordi La Forge and Data went awry and influenced the progam, causing the main characters to look like Data and even removing the holodeck safeguards, making the 19th century firearms deadly. (TNG:"A Fistful of Datas")