Hi Espruino Guys!
Then I thought to leave it standalone, and connect to it wirelessly. The Web IDE connected and started working, but suddenly it dropped the connection after 20 seconds. A reboot reconnected, only to have dropped it again. This behaviour (connection dropping in under 1 minute) was "consistent".
First I suspected the IDE client, so I just powered the board - having Wifi connection on boot - and started to ping it from outside. And I discovered that the Wifi connection itself was dropping!
Now I suspected the power supply, so I put a 3A one, then the Wifi signal strength. So I moved the board closer to the router - it reported an rssi of -55 dBm, which should be very good. Unfortunately neither solution did help in any way.
So I did not know what to do. Lastly, I flashed a Micropython firmware, which did not show any instability with the Wifi. Sorry for it.