|The last thing you want to do when your computer even MIGHT shut down is try to update your BIOS. This is because if your issue was actually an overheating issue or a hardware issue then it would definitely happen during the update and almost certainly then become a very expensive paperweight. If there is any way to correct this, Dell is the one that might have a way to do it.|
"I know for sure that this is not an issue of overheating as the laptop was not actually hot." - How do you 'know' this? If your system/CPU is overheating, especially if it is quickly, this would not become apparent from the outside. Running a program like HWMonitor or going into your BIOS and monitoring your temperature from there, for maybe an hour or more might tell you if it is a heat issue.
Had you come here first, we would have recommended testing your internal temperatures and some hardware tests (memory and hard drive to start with), not a BIOS update.
If the computer is actually new and under warranty, you might have an small possibility of a replacement (more likely if they were the ones who told you to do the BIOS update), if it is not under warranty, especially if you 'recently acquired' it from someone else as a used machine, you may get some support from them (you might have to pay for it), but you will not get a replacement machine.
Sorry if this is not what you want to hear, but you may have just made an expensive mistake.
You have to be a little bit crazy to keep you from going insane.