This might sound like a power management problem - I remember facing this kind of issue with USB devices lost because of poor power delivery.
I could fix this by unchecking the power management button.
So, I did a quick search and found that for linux the ethernet power management is located in /sys/class/net/eth0/device/power, where the file control contains auto if power management is allowed, and on otherwise.
Try sudo nano /sys/class/net/eth0/device/power/control and replace auto with on, then Ctrl+O, Ctrl+X, or (easier) just do a copy/paste of the command given there , reboot and... keep your fingers crossed.
Let's know if it works.

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