The thing to keep in mind is that throughout most of human history, playing dead was terrible idea. Reason-being, up until the 20th century, looting corpses was generally considered legal foraging. (Up until the 20th century, pillage was frequently a stated incentive for a soldier.) Once the losing side was driven off, the fallen would be stripped naked by the victors and their assorted camp followers. Also worth mentioning, ripping teeth from the fallen was a common practice up through the Victorian-era, as they could be resold for dentures. Good luck playing dead through that. Unless you're a wounded officer/noble worth a ransom, there was no incentive not to finish you on the spot.
That being said, I can think of some modern US examples.
The D-Day paratrooper on the St. Mere Church is a pretty famous story, immortalized in the book/movie The Longest Day. The planes took so much AA that the paratroopers were scattered everywhere from early drops and evasive manuevers. Two squads ended up dropping literally in the middle of the German occupied town of St. Mere Eglise. Private John Steele's chute got caught on the church steeple. When he attempted to cut the risers, he fumbled and dropped his switchblade. His weapon would have been in an inaccessible duffle bag. So he did the only thing he could and played dead, having to watch the rest of his squad get hunted down and killed.
The classic book The Longest Day was written as pop history and is a very fun read. The movie also has aged fairly well, so long as you're comfortable with how WW2 was depicted on the silver screen prior to Saving Private Ryan. The 505th RCT also maintains a pretty good history page that does PVT Steele good justice. TIL, that Steele was actually the oldest man in his company at 32 and doubled as the company barber: http://www.505rct.org/album2/steele_j.asp
In Vietnam, there were quite a few lone suvivors of catastrophic near ambushes.
In OPERATION HICKORY, 18 May 1967, an infantry platoon was pinned by a battallion-sized element of NVA. Realizing they where doomed, the PL called 155mm on their position. The NVA overran them and took their position. When relief came the next morning, they found only eight survivors who had successfully played dead and were stripped of belongings. It's worth mentioning that this action was the beginning of a Presidential Unit Citation for 1/4 BCT.
http://members.tripod.com/msg_fisher/battle-1a.html http://members.tripod.com/msg_fisher/puc.html
The POW's SPC McMillian, SGT Davis, and SGT Calloway were captured while unsuccesfully attempting to play dead. Their accounts were told in "Survivors" by Zarin Grant. On 10 MAR 1968, their observation post came under fire and the did not have a radio. While evading incoming mortar fire, they were separated from their unit and walked into the line of an enemy machinegun. Calloway was badly wounded and they tried to hide. Eventually they exhausted their ammo and were captured while trying to play dead. Luckily the NVA had recent order to take POW's as negotiating chips. Calloway died of wounds that night. Davis and McMillan spent the next five years getting tortured in a variety of POW camps in both North and South Vietnam. On a positive note, Davis would retire as a Command Sergeant Major in 1997 (!!!) after finishing a 30 year career with the US Army.
http://www.pownetwork.org/bios/d/d016.htm http://www.veterantributes.org/TributeDetail.php?recordID=1716
There are various loose anecdotes about this being done in Vietnam. Particularly chilling was a 2010 obit for a deceased Marine vet:
"As a Marine on the front lines, Mr. Cruse saw most of the men in his platoon killed, his family said.
Viet Cong ambushed Mr. Cruse and men in his division as they crossed a mine field. Mr. Cruse took shelter behind the body of a Viet Cong soldier who had just been killed, said nephew Jeffrey Dovan, who is now a Marine.
When the gunfire stopped and the Viet Cong were checking for survivors, Mr. Cruse successfully played dead.
Mr. Cruse, who lived in Vietnamese villages and dressed like the local people to blend in, later had a difficult time coping with what he had experienced.
"He had issues with helicopters," his sister said. "It was bad for a long time, with the flashbacks.""
http://articles.philly.com/2010-09-14/news/24974768_1_sister-viet-cong-vietnam-war
One unusual outlier, on Christmas, 1969, CPT Marshall and three other men were enroute to a Christmas party outside Saigon when their jeep was ambushed. The only man alive, he held his breath as they looted his corpse. The dog piled the bodies and lit them ablaze with gasoline. Luckily Charlie did't watch the bonfire as CPT Marshall was able to crawl out from the bottom. "I could smell the burning flesh."