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.

Ok so thanks to this excellent video from i80386sx I was able to bring that little machine back to life !!

The design of the machine is actually quite clever, it's just a pity that to access the coin cell battery man needs to disassemble the whole thing!

It's a slightly beefed up configuration with 6 MB RAM and a Conner 80 MB hard drive.

It even has a licensed copy of McAfee 😀 although the virus definition files are slightly outdated (1997)...

I am amazed to see that Windows 3.1 is able to guess that "24" means 2024...

Nothing really interesting on the disk. Most of the files are dated of 1993/94, and surprisingly a few newer documents dated of 98/99... I am surprised to see that out of the 80 MB there's only 12 MB free, with only Windows 3.1, MS-DOs 6.2, Excel and Word installed and only a bunch of Word and Excel documents.

Anyway I am super happy to have revived that little Compaq, it'd been a while I hadn't played with Windows 3.1 ! A big thank you to Thermalwrong for pointing me in the right direction. All it took was to change the RTC battery.

Btw does someone know where to find the y2k patch for Windows 3.1 in french?

EDIT: Wow it is almost impossible to find. I believe I manage to locate here: http://lavenu.x.free.fr/cours/downloads.htm but I need to test to confirm.

Wow the screen on that one looks great!
I'm glad it was just the RTC battery that was stopping yours from working the same as it now does mine - I got a 4/25E which was a 486 with the active matrix monochrome display, but the only replacement board I could get was the 3/25 so now it's a 386 😀 The active matrix monochrome screens seem to often be suffering from 'tunnel vision' where the corners black out like vignette on a photograph, it doesn't seem like the passive mono/colour LCDs have that problem.
My solution instead of using the original battery was to bodge a CR2032 onto there but I found it ran down pretty quick so now it's got a battery tray in the side and I just install the battery when I want to use the laptop.

One thing I've read as well is that replacing the hard drive with something like a CF card can be hazardous so keep using the original drive.

Interesting, I did not know you could swap the mobo between the two models, the case of the 4/25E looks a bit different from the Lite 25. Does the color screen works with the 386?

I am indeed happy with the screen on mine, it's pretty good and there's no tunnel effect at all. It reminded me of some old Apple Powerbook on which I had to work when I was student, and in fact it's probably the same screen as the Wikipedia page for the LTE Lite indicates.

Honestly the only regret are the cracks on the hinges and case, which are quite bad. A piece at the lower back corner came out during the disassembly.