The Wien Mitte CAT terminal in Vienna has a bag drop, but their website isn't clear about which airlines it accepts baggage for. I know you can drop bags for Austrian Airlines, because I've done it.
Can I do the same for British Airways?
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As of the date of my answer you can check in for Austrian, Lufthansa, Eurowings, Swiss and Brussels Airlines, at the railway station.
The full details of check in and sevices are detailed on the service specific website
It is worth noting that when I tried using this service I was traveling on a codeshare and had a United Airlines ticket, but was flying with Lufthansa. They had difficulties in checking me in and I had to go to the airport to do this. The staff at the railway station said that this was uncommon, and I should have been able to check in.