What happened to Gandalf the Grey's body when he was sent back as Gandalf the White?

So Gandalf and the Balrog of Moria have a little fisticuffs and they both died. Presumably dead bodies lie on the mountain side somewhere.

Gandalf "strayed out of thought and time", gets a can of spinach and a new colour then he comes back sometime after his death as Gandalf 2.0.

The questions are thus:

1. Did Gandalf's old body get burnt to a crisp, or suffer some other death that meant there was no dead body left? OR


2. Did his old body get renewed?

Or do we not know? If his body was left in the snow, then wouldn't it have been preserved due to freezing? This seems like an obvious oversight in the story.

I'd like to stick with the book canon, but out-of universe answers are also welcome.

Clarification: I'm not asking about his soul or whether he can die or where he went. I am specifically asking about what was left behind on the mountain.