While we can use pip directly as a Python module (the recommended way):

python -m pip --version

This is how I installed it (so it can be called directly):
Firstly, make sure that command pip is available and it isn't being used by pip for Python 2.7

sudo apt remove python-pip

Now if you write pip in the Terminal, you'll get that nothing is installed there:

pip --version

Output:

Command 'pip' not found, but can be installed with:

sudo apt install python-pip

Install python3.8 and setup up correct version on python command using update-alternatives (as done in the question).

Make sure, you have python3-pip installed:
(This won't work without python3-pip. Although this will install pip 9.0.1 from python 3.6, we'll need it.)

sudo apt install python3-pip

This will install pip 9.0.1 as pip3:

pip3 --version

Output:

pip 9.0.1 from /usr/lib/python3/dist-packages (python 3.6)

Now, to install pip for Python 3.8, I used pip by calling it as a python module (ironic!):

python -m pip install pip

Output:

Collecting pip
  Downloading https://files.pythonhosted.org/packages/36/74/38c2410d688ac7b48afa07d413674afc1f903c1c1f854de51dc8eb2367a5/pip-20.2-py2.py3-none-any.whl (1.5MB)
  100% |████████████████████████████████| 1.5MB 288kB/s
Installing collected packages: pip
Successfully installed pip-20.2

It looks like, when I called pip (which was installed for Python 3.6, BTW) as a module of Python 3.8, and installed pip, it actually worked.

Now, make sure your ~/.local/bin directory is set in PATH environment variable:
Open ~/.bashrc using your favourite editor (if you're using zsh, replace .bashrc with .zshrc)

nano ~/.bashrc

And paste the following at the end of the file

# set PATH so it includes user's private bin if it exists
if [ -d "$HOME/.local/bin" ] ; then
    PATH="$HOME/.local/bin:$PATH"
fi

Finally, source your .bashrc (or restart the Terminal window):

source ~/.bashrc

Now if you try running pip directly it'll give you the correct version:

pip --version

Output:

pip 20.2 from /home/qumber/.local/lib/python3.8/site-packages/pip (python 3.8)

Sweet!

Another solution would be to install the pip that is in apt. sudo apt install python3-pip. The version of the pip that it installs is for all versions of Python not only for version 3.6 once installed you just need to update the pip with the command python3.8 -m pip install pip and he will be install the latest version of pip for Python.

I would not advise you to remove Python2 because it is an important module for the system you should just create a permanent "alias" in .bashrc for Python3 I did like this alias python="python3.8.

I did this a couple days ago and I struggled a lot with it but I finally got it working, so I wrote up what I did as a blog post.

In the end I think I may have done mostly the same things as the above answer, but if you got lost following it, maybe my screenshots etc will help.

Here's the tl;dr of the process I did:

• Uninstall python3-pip & python-pip using apt
• Remove the old pip files from /usr/local/bin
• Reinstall python3-pip using apt
• Add $HOME/.local/bin to your $PATH (also restart your shell to make sure you did this right)

# install py3.8 and dependencies for the pip3 bootstrap script
add-apt-repository -y ppa:deadsnakes/ppa && \
    apt install -y python3.8 python3.8-distutils

# download and run the pip3 bootstrap script
cd /tmp && wget https://bootstrap.pypa.io/get-pip.py && \
    python3.8 /tmp/get-pip.py

# use pip py3.8 module to install python packages
python3.8 -m pip install numpy pandas

As suggested in official documentation you can try with get-pip.py.

wget https://bootstrap.pypa.io/get-pip.py
python3.8 get-pip.py

This will install pip as pip3.8