Is there any well documented case where either the attackers or the defenders relied more or less exclusively on .22 ammunition.
To be clear, you're asking about .22 rimfire, correct? Because an M16 fires the NATO standard 5.56 which is a .22 caliber round. The .223 Remington and 5.56 are interchangeable as long as the rifle can accept the pressure, .223 having a higher pressure.
If you're only asking about the rimfire, then I don't know. If you're asking about all .22 caliber rounds, then the answer is yes - every conflict involving US or Soviet forces since the mid 60's (US) and 70's (USSR). The M16 and the AK74 are both chambered in .22 caliber rounds.
Historically, rimfire cartridges HAVE been used in war, but in larger calibers - between .30 and .58. In .22 caliber, the old .22 Flobert round has enough power to get stuck in your skin or put your eye out but not much more. .22 shorts and .22 longs are respectively stronger, and the .22 long rifle and the .22 winchester magnum rifle (the only 2 commonly remaining cartridges left) are strong enough for pest control ~ rabbits, gophers, rats. A .22lr is quite dangerous. It has enough power to penetrate a human skull but generally not enough power to exit the skull. Instead, it fractures and destroys the brain, leaving lead pieces everywhere inside the skull. Ironically, because of the lack of recoil and small rifles, this is usually a gun given to kids as their first gun, but nake no mistake, they are quite dangerous.
As to the original question, I know of no battles involving .22 rimfire rounds. It was never a round issued in any great quantities to military forces (CIA and other special ops have silent .22 lr pistols) and even the most powerful .22 WMR round can be stopped by a textbook. That said, I can source everything else EXCEPT that statement.
EDIT: Here are a couple of links about .22 LR pistols in Vietnam.