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In the eighth episode of the first season, Jon Snow and Jeor Mormont come across two dead rangers. The rangers were dead for weeks and once accompanied Jon's uncle, Benjen Stark.
After Jon and Jeor bring the men's bodies back to Castle Black, one of the bodies reanimates into an ice zombie. Jon runs it through with a sword, but that eventually destroys it with fire.
Sam read that corpses touched by the White Walkers are reanimated and can only be killed by fire.
Why weren't the corpses converted to wights minutes after being killed by White Walkers? Why did it take them weeks to reanimate?
Edit to add: The other question linked to this one asks if dead bodies can become wights south of the wall, while this one asks if they become wights weeks after White Walkers touch them. That one is about the location of becoming a wight, and this one is about time since being killed and transformed by a White Walker. Close, but not a duplicate. :-)
Please give citations from the books if you can. I consider the books a higher canon than the TV series.