I have just had exactly the same problem (in 2019), on ubuntu 19.10, while copying large number of files from USB disk to the SATA disk. Both filesystems are ext4. When I turned off the swap, the problem disappeared. It looks like some bug in memory allocation for disk buffers - apparently, the kernel tries to allocate as much memory for disk buffers, as possible in such situation, which doesn't make sense (making disk buffers in swap...), or it just wrongly calculates the memory size than can be used for caching...
As someone pointed out, setting swappiness to 1 doesn't solve the problem, which is logical, if the files you copy have total size more, than the RAM size...
By the way - can anybody explain, WHY turning off the swap is not recommended? If I have 32GB of RAM, what difference does it make to add another 4GB of swap? I can only think of some obscure applications that actually expect the swap to be there. But I would just stop using such application, since properly written one shouldn't care about swap. The swap should only be managed on the OS level...