Cannot install pip module because there is 'no space' left on device

When I try to install the pytorch module for python3.8 pip complains that there is no space left on the device e.g.

$ pip3 install torch
Collecting torch
  Using cached torch-1.8.1-cp38-cp38-manylinux1_x86_64.whl (804.1 MB)
ERROR: Could not install packages due to an EnvironmentError: [Errno 28] No space left on device

But according to df there should be enough space around both on the system partition as well as in tmpfs.

Output from df:

$ df -h
Filesystem      Size  Used Avail Use% Mounted on
udev            7.7G     0  7.7G   0% /dev
tmpfs           1.6G   20M  1.6G   2% /run
/dev/nvme0n1p4   23G   20G  2.3G  90% /
tmpfs           7.7G  127M  7.6G   2% /dev/shm
tmpfs           5.0M  4.0K  5.0M   1% /run/lock
tmpfs           7.7G     0  7.7G   0% /sys/fs/cgroup
/dev/nvme0n1p5  183G   48G  126G  28% /home
/dev/nvme0n1p1  256M   37M  220M  15% /boot/efi
tmpfs           1.6G  5.8M  1.6G   1% /run/user/1000

I did try to use a different TMP_DIR as suggested by some StackOverflow answers like this one but that did not fix the problem in my case. My system drive seems to be a bit full but it looks like there should be enough space to install the package. I also tried to install the package locally (e.g. pip3 install --user) but that did not help either.

Pip worked for all the other packages I tried to install but for this one. Though the other packages were notably smaller.

Ubuntu version: 20.04.1 LTS

Pip version: 20.0.2 for python 3.8

The most likely cause is lack of space in /tmp. The workaround is to instruct pip to use an alternative folder. I use the following formulation:

TMPDIR=/home/user/tmp/ python3 -m pip install a_package

I tried using the cache-dir argument but it fails with a "file not found" error.