It's not impossible, but it would be a poor decision if you wanted to have a good chance of survival. The dead and wounded were not sacrosanct on the medieval battlefield. Soldiers would pick through bodies to loot the dead or find wounded who could be captured and ransomed. Quite often, wounded soldiers would be finished off with a dagger to the throat if they were not identified as important/valuable enough to bother capturing, or in cases of civil war where emotions were running even higher than normal on the battlefield. Even if one could play dead through the entire battle and happened to not be discovered by a victorious enemy (or unscrupulous ally), there would still be a huge amount of risk as local civilians would descend on the battlefield like vultures to pick over the scraps. These locals have just experienced the traumatic and destructive experience of having two armies march around the area and will not be in a mood to be sympathetic to soldiers of either side. Playing dead might leave someone more vulnerable than simply running away, which was the usual option of medieval soldiers when they perceived that the battle was not going in their favor.

As a follow up question, what would happen to you if it was discovered that you were playing dead?