So, I do not know how much junk you do have on your SSD, but in most cases when regarding SSD Disks, the recommended space to have with absolute nothing is and can be up to around 100 and even plus GBs.

Try resetting your console, and see if it happens within an hour of gameplay, if you are playing for an extended period of time much like PC's and depending on the game the console can get sluggish. A occasional restart after a few hours will do wonders.

You can also check if your consoles fans/exhaust are dirty, it can be getting hot.

I don't know if consoles have a tool in system options to check drive for errors, if it does try running that. 

If there are no errors it might be a capacity issue like Kundari says.

If all this fails, then its probably a nurgling living under youre bed, that is sabotaging your fight against the void.

I don´t think and will not imagine it being the Server´s fault though.

Could it be your SSD´s age? Because I know and also heard that SSD is not the same thing as the usual old regular Harddisk is.