After thoroughly inspecting the motherboard, I found a component around the Marvell chip that has partly come loose.
The component description says "CE5"
After a re-soldering, the device works again.
Now the NIC boot looks like this:
[ 26.221643] Intel(R) Gigabit Ethernet Network Driver - version 5.3.5.3
[ 26.228957] Copyright (c) 2007-2015 Intel Corporation.
[ 27.588010] igb 0000:00:14.0: added PHC on eth0
[ 27.593104] igb 0000:00:14.0: Intel(R) Gigabit Ethernet Network Connection
[ 27.600815] igb 0000:00:14.0: eth0: (PCIe:integrated:integrated)
[ 27.607667] igb 0000:00:14.0 eth0: MAC: 00:a0:c9:00:00:00
[ 27.613793] igb 0000:00:14.0: eth0: PBA No: 002100-000
[ 27.620377] igb 0000:00:14.0: LRO is disabled
[ 27.625257] igb 0000:00:14.0: Using MSI-X interrupts. 1 rx queue(s), 1 tx queue(s)
[ 28.986059] igb 0000:00:14.1: added PHC on eth1
[ 28.991141] igb 0000:00:14.1: Intel(R) Gigabit Ethernet Network Connection
[ 28.998899] igb 0000:00:14.1: eth1: (PCIe:integrated:integrated)
[ 29.005727] igb 0000:00:14.1 eth1: MAC: 00:a0:c9:00:00:01
[ 29.011858] igb 0000:00:14.1: eth1: PBA No: 002100-000
[ 29.018457] igb 0000:00:14.1: LRO is disabled
[ 29.023330] igb 0000:00:14.1: Using MSI-X interrupts. 1 rx queue(s), 1 tx queue(s)
[ 30.385164] igb 0000:00:14.2: added PHC on eth2
[ 30.390255] igb 0000:00:14.2: Intel(R) Gigabit Ethernet Network Connection
[ 30.397964] igb 0000:00:14.2: eth2: (PCIe:integrated:integrated)
[ 30.404797] igb 0000:00:14.2 eth2: MAC: 00:a0:c9:00:00:02
[ 30.410922] igb 0000:00:14.2: eth2: PBA No: 002100-000
[ 30.417495] igb 0000:00:14.2: LRO is disabled
[ 30.422374] igb 0000:00:14.2: Using MSI-X interrupts. 1 rx queue(s), 1 tx queue(s)
[ 31.783311] igb 0000:00:14.3: added PHC on eth3
[ 31.788409] igb 0000:00:14.3: Intel(R) Gigabit Ethernet Network Connection
[ 31.796115] igb 0000:00:14.3: eth3: (PCIe:integrated:integrated)
[ 31.802940] igb 0000:00:14.3 eth3: MAC: 00:a0:c9:00:00:03
[ 31.809058] igb 0000:00:14.3: eth3: PBA No: 002100-000
[ 31.815634] igb 0000:00:14.3: LRO is disabled
[ 31.820510] igb 0000:00:14.3: Using MSI-X interrupts. 1 rx queue(s), 1 tx queue(s)