If the POST card reset LED stays on then it's the reset circuitry issue. I would recheck the battery damaged area for broken traces.

Did the keyboard work on yours? I have the same board as you (but with a socket instead of soldered CPU) and I've been bashing my head back and forth getting the KB to work. Same exact issue - corrosion from VARTA battery, but mine did POST (came with a TI 486DLC) and stop with a Keyboard error, even after redoing most of the traces on the bottom with wires (since my soldering iron wasn't up to snuff for doing trace work) it still complains about no keyboard.

And to answer your question, yes, it seems it's possible to replace the soldered CPU. Checking against mine, you'd just need to remove the soldered CPU and add a socket, since the needed components for the socket are already there.

Atlantis wrote on 2020-07-10, 04:14:

Few wire bridges had been soldered and instead of original VARTA battery there was a small li-pol accumulator! I removed it and replaced with standard CR2032 with diode. I cleaned battery area with an IPA cleaner.

wouldn't have needed to replace the battery, the connector right below it is a connector (J6) for external battery 😉

https://www.arvutimuuseum.ee/th99/m/S-T/32636.htm
please disregard the cache placement in TH99 it is wrong. yours is already correct.

Thanks for all the suggestions. I removed some elements nearby the CMOS battery. There are in fact some signs of corrosion. I will try to clean it with vinegar and then IPA, check continuities and tin traces.
Is there a way to fin a trace that is most probably responsible for this issue? Is there high possibility if rescuing that board?
By the way, someone needs BIOS image? EPROM was soldered, but i removed it due to inspection and made an backup image. Now of course I will socket it.

There is a 386 BIOS image collection thread if you like to upload your BIOS image. As for the traces, my suggestion is to check all traces near the corroded area. But if you would like to start with the reset line, you can start tracing it from the ISA slot, pin B2. Please see ISA pinout here. Don't forget to check the traces underneath the discreet components too, I usually lift just one leg to clean and check underneath, so I don't have to remember which component is at what place. 😀

I think you should have no problem saving this board, the corrosion is largely localized near the battery area, so with some good cleaning and connecting back the traces, it should be fine.

ISA RESET signal (B2 pin) should been connected directly to the CPU/Chipset RESET lines? It isn't.
I also assume LT38C41 is strict, pin compatible equivalent of 8042/8242 chip and its 4th pin is RESET signal? There is no continuity, but it makes sense as it is active low.

Not sure about 486s, but usually CPU's reset line is generated by the chipset. For example, here is one for the Pentium 4. The reset button/circuitry sends a signal to the south bridge, south bridge relays the reset to the I/O slots and north bridge, and the north bridge sends a reset to CPU. Since your POST card's reset LED stays on, I'm suspecting there are some broken traces or short circuit caused by the corrosion, since pin B2 is very close to the end of the ISA slot.

UPDATE: I found and patched two more broken traces leading to first cache chip. It didn't help.
Any hints how to locate broken reset trace? 😀

EDIT: Board works. It turned out to be a power supply issue.
I spotted some other problems while testing this board with Checkit. Board restarts itself after executing memory test, but only with certain SIMM sticks. Does it mean those memory modules are broken or just somehow incompatible?
How long does it take to test entire (8MB or 16MB) memory ?

At least on mine, the RAM count goes pretty fast (~2 or 3 seconds?) before throwing the keyboard error.
(not sure if it's the KB chip being bad, or I didn't trace my wires correctly.)

Ok, I was able to run memory test with few different programs. It turned out to be a protected mode fault.
Now I noticed another issue. It looks like there was no cache at all. None of the programs is able to see it: I tried Checkit, CacheCheck and MemTest86+ 2.11..
I also can't locate cache related options (enabling/disabling) in BIOS.
I've heard about infamous PCChips 486 boards with fake/inactive cache chips. Were there also such 386 devices?
This would be a good explaination for wierd observation I made - there were two broken traces leading to one of the cache chips (effect of battery acid leak) but board had been working. Nothing had changed after I patched them up.

Is it likely that cache is not detected due to battery leak related damage, which I wasn't able to locate, or was it always that way and this cache is "fake"?

EDIT: Ok, now i see there are cache enabling disabling options in BIOS. But internal cache is permanently disabled and changing external cache setting doesn't seem to have any effect.