Adding this as an additional answer, since it technically differs from my Gun Fight answer.
mnem mentioned the Fairchild F8 microprocessor in the comments of my question. If we are indeed counting the Fairchild F8 as an 8-bit microprocessor then Gun Fight is technically not the first arcade game ever created using a microprocessor. It is, however, the first commercially released game that used a microprocessor.
Jerry Lawson, designer of the Fairchild Channel F video game console, allegedly created a top-down driving coin-op game using the Fairchild F8 in 1974/75.1
The Golden Age Arcade Historian comments:
Sidebar - Was Demolition Derby the first coin-op game with a microprocessor?
Some sources have suggested that Demolition Derby was the first game
to use a microprocessor and even that the game was released not long
after Pong, but is this true? Lawson claims he started working on the
game in 1972 or 1973 and sold it to Major Manufacturers of San Mateo,
CA. Some sources (including the Wikipedia article on Lawson) claim
that the game "debuted" shortly after the release of Pong. The F-8,
however, was not released until 1975 and Major Manufacturers was not
incorporated until October of 1974. The October, 1975 issue of Play
Meter announced that at the 1975 MOA show (the same show where Gun
Fight was introduced), Major Manufacturers would be "…. introducing
two new upright games that use a microprocessor...instead of a logic
board, as well as exhibiting their line of video games and a new
designer cocktail table". The article does not name any of these
games, nor do any other issues of Replay or Play Meter. It is not
clear from the description if the microprocessor games were video
games or not. The October, 1975 issue of Vending Times, however, does
list two games that the company was to display at the MOA: Lunar
Module and Fascination - but does not mention whether they use a
microprocessor. The 1972/1973 date thus seems clearly too early, at
least for a microprocessor version of Demolition Derby (though Lawson
could have started with a non-microprocessor version). In addition,
only one copy of Demolition Derby is thought to have been built and it
never went past the field testing stage. On the other hand, while it
seems unlikely that it was field tested prior to 1975, given that
Major Manufacturers did plan to show microprocessor games at the 1975
MOA, it (or one of Major's other games) may have been tested prior to
the release of Gun Fight.