The first guns with multichambered cylinders that revolved to feed one barrel were made in the late 1500s in Europe. They were expensive and rare curiosities. Not until the 1800s would revolvers become practical weapons for non-rich owners. One of the first was a flintlock revolver made by Elisha Collier in 1814. The first percussion cap revolver was the Colt Paterson, invented by Samuel Colt in 1836.[1] The first cartridge revolvers were produced around 1856 by Smith & Wesson.

Blue,

I know of numerous flintlock "repeating" firearms, one as early as 1750...but these were civilian sporting or defense arms.

The first repeating rifle adopted by the military THAT I AM AWARE OF was Colt's Model 1855 revolving rifle. It was adopted for trial by a small number of US Army units and saw action in some of the skirmishes with Indians and limited use in the Civil War.
I believe Berdan's sharpshooters used this rifle while waiting for the arrival of their breech-loaders.

This rifle loaded just like the cap and ball revolvers, and, like those pistols, would ocasionally discharge more than one chamber at once. Because it was a rifle and one of the shooter's hands was supporting the barrel ahead of the cylinder, a multiple discharge usually meant some fingers were lost. Because of this, and it's slow loading, it did not find great favor with the military and the US Government eventually sold them off for 42 cents each just to be rid of them.

Col Murph or someone may know of an earlier military repeater.