Indeed, Luther did claim "I was a good monk, and I kept the rule of my order so strictly that I may say that if ever a monk got to heaven by his monkery it was I. All my brothers in the monastery who knew me will bear me out. If I had kept on any longer, I should have killed myself with vigils, prayers, reading, and other work." So I started digging around a bit, and while I couldn't find any actual helpful sources to document this fact, I found the following versions of this tale:
"As a young monk Luther was obsessed with atoning for his sins and went through vast lengths to punish himself. This ranged from extreme self denial and physical and mental tests to self flagellation. One such punishment consisted of lying in the snow, through the night at the height of winter until he would have to be carried back inside" [source] [source].
"These included fasting, manual labour and could include sleeping on hard benches, without any blankets on or in Luther's case sleeping or lying out in the snow" [source]
"Did you know...that as a young monk in Erfurt Luther was obsessed with atoning for his sins and went to ridiculous lengths to punish himself. This ranged from extreme self-denial and self- flagellation. One night, at the height of winter, he was found lying in the snow and had to be carried back inside the convent" [quote]
"In the monastery Luther was scrupulously obedient. He fasted. He prayed. He was diligent in his work. He sometimes slept in the snow without a blanket in order to mortify his flesh, so that he might be more pleasing to God" [source]
Now there may indeed be some sort of documentation for this tale, but in the 30 minutes or so I looked for it, I came up with nothing. If it does exist, my guess would be it's a Table talk statement. On the other hand, that some of the main biographies of Luther I checked don't mention it, this makes me just a bit suspicious that perhaps it's yet another Internet myth. That the story has multiple versions also is suspicious.
If anyone has any sort of documentation, please leave a comment.