the lord of the rings - 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.

Answer

The book suggests that his spirit reinhabited his original body, and there seems no reason to suspect otherwise.

First let's look at the deaths of both Gandalf and the Balrog (quotes from the Two Towers chapter the White Rider):

I threw down my enemy, and he fell from the high place and broke the mountain-side where he smote it in his ruin. Then darkness took me; and I strayed out of thought and time, and I wandered far on roads that I will not tell.

Then Gandalf's resurrection:

Naked I was sent back – for a brief time, until my task is done. And naked I lay upon the mountain-top. The tower behind was crumbled into dust, the window gone; the ruined stair was choked with burned and broken stone. I was alone, forgotten, without escape upon the hard horn of the world.

This makes it clear that he was resurrected in the same place where he had died, but - of course - his body could have been remade. However, and a little further on in the text, we learn that after Gwaihir brought him to Lórien:

I tarried there in the ageless time of that land where days bring healing not decay. Healing I found, and I was clothed in white.

Evidently therefore his body needed healing, which we can reasonably suppose it would not have if it was a new body. Hence we can deduce that it was his original body, not renewed.

If we cross-check this with Letter 156 we see confirmation of the "Gandalf needed healing" hypothesis:

Galadriel's power is not divine, and his healing in Lórien is meant to be no more than physical healing and refreshment.

So there is no oversight in the story; Gandalf evidently just reinhabited his old body, which then needed to be healed of it's wounds in Lórien.