The Warrior Monk is a unit in Civilization VI, introduced in the Fall 2017 Update. It can only be purchased with Faith in a city that has a majority religion and a Holy Site with a Temple (or one of its replacements).
The Warrior Monk is a military unit with its own special set of Promotion: Cobra Strike, Dancing Crane, Disciples, Exploding Palms, Shadow Strike, Sweeping Wind, and Twilight Veil. A fully promoted Warrior Monk will have 65 Combat Strength and 4 Movement, be able to attack twice per turn with double flanking bonuses, be invisible to non-adjacent enemy units, and spread its civilization's religion to nearby cities each time it defeats an enemy unit.
There are, however, a number of factors that make the odds of a Warrior Monk earning all of these Promotions rather slim. The timing of this unit is very poor, rendering its presence in battle a rare sight; a domination-religious strategy that heavily involves Warrior Monks is unheard of. Practically, you can get the Warrior Monks Follower Belief when you found your religion, but you cannot purchase the unit until you have a Temple, which is unlocked with Theology. Considering that Science output naturally outpaces Culture output in the first two eras, Theology has a lot more prerequisites than Iron Working, and the Inspirations leading to Theology are so much harder to trigger than Eurekas leading to Iron Working, unlocking Theology before other civilizations have access to Swordsmen is impossible, so the landscape of the game is always against the Warrior Monks. Even in Gathering Storm, any Warrior Monk purchased in a city to which Moksha has been assigned after earning the Patron Saint title will receive a free Promotion, but even that does not save this unit. Other melee units can fight and earn experience from the beginning of the game, have unique bonuses and buildings to boost their experience gain, and have Tier I Promotions that enhance their sturdiness or damage output in battle, whereas the Warrior Monk has none. Buying just 2 Warrior Monks means you have to pass up roughly 4 Missionaries (they have scaling Faith cost) or an Apostle, both of which are more useful than a unit that is so weak in battle without Promotions. Overall, there are always better Follower Beliefs to choose and many other better ways to invest your Faith.
Although they are melee attackers, Warrior Monks are not classified as melee units. They don't benefit from Battering Rams or Akkad's Suzerain bonus, receive neither Combat Strength modifiers from Fascism, Digital Democracy, and Ambiorix's ability nor bonus XP from Encampment buildings, and aren't boosted by Great Generals of any era. They do, however, benefit from Siege Towers, and also receive the bonuses from Oligarchy and Comandantes Generales.
The myths of wall-running, magic-palmed warrior monks endure because of how outlandish they are. Unfortunately, we simply have to live with the actual feats of history's priests with astounding fighting skills.
Consider the Shaolin Monastery in China, founded by the Indian Buddhist monk Batuo in 495 CE. While the history of martial training at Shaolin is tangled with legends and pop culture, it's said that Batuo and his first Chinese disciples, potentially soldiers before joining the monastery, already came equipped with martial arts skills, passing them down to their students.
By 581, monks at Shaolin would have their own fighting style, even siding with would-be second Tang Emperor Li Shimin during the dynastic wars of the period.
By contrast, Japan's Buddhist sohei would engage in the less glamourous inter-temple turf wars, picking up arms and using their fighting skills in order to gain political, not spiritual, dominance.