The 386 model can not power up if the RTC battery is dead. To change that you'll need to take it apart and pull out the systemboard which is in a big aluminium shell.

The top case comes off and upwards by prying some plastic clips around the sides and maybe some screws in back? Not sure, mines in storage right now. It's not easy to do but if you use a plastic pry tool / guitar pick around the seam from the side and pushing upwards the clips should release.
i80386sx is the expert on these - check out his channel: https://www.youtube.com/@i80386sx/videos
And specifically this video: https://www.youtube.com/watch?v=hL2myu5qjXQ

Once you've got the top case off then make sure to gently remove the flex cables as they are basically irreplaceable.
Once the keyboard and hard drive and screen are out / off then you can pull the motherboard's aluminium shell out of the lower plastic case.

Since it's the 386 model and not the 486, it's more likely that just the RTC battery being dead is stopping the computer from working. Replacing that should allow the laptop to work again.
Personally I ran wires for the RTC battery with a CR2032 cell holder into the expansion bay so I can replace it / take it out for storage without having to disassemble the machine anymore.

Check whether there's a NIMH cell inside the motherboard shell since that destroyed my 486 'LTE Lite 4/25E' motherboard. I'm not sure the 386 models are so at risk from that, you might be lucky.

Thanks a lot, I'll try to change the RTC battery and see if this helps.