Until a few weeks ago, the latest HPLIP driver version, available at HP Developers - HP Linux Imaging and Printing, was 3.18.12. At Ubuntu 16.04 official repository, the available version was 3.16.3.
First I tried to install the latest version, downloaded from the website. However, a compatibility issue raised with the package
python-gobject. As noted by N0rbert (main answer), you need to completely uninstall the actual version of HPLIP before you install another version. Do it manually, as described by N0rbert. After that, ensure that the following files are not present any more (or that the respective folders are empty):
I am not sure if all the files listed above absolutely need to be removed. However, with them gone, I could proceed with the new installation without problems.
Although the repo doesn't have the latest version of the driver, I decided to install it, via
apt, because it works without crashing with other preinstalled packages. I did as follows:
sudo apt install hplip (Linux driver for most HP's printers and scanners).
sudo apt install hplip-gui (required by hp-plugin).
hp-plugin (as normal user).
This plugin is a proprietary "part" of the driver which HP doesn't open-source. The device won't work without it. You will be prompted to accept the terms and install the required plugin for the exact version of the driver you have installed.
hp-doctor (to ensure everything is installed properly, especially the plugin).
hp-setup to add the connected device, or use the HP's software (with graphical interface), already installed at this point.
Go to Ubuntu System Settings and add a new printer/scanner with HPLIP.
PS: By January 2019, the latest version is 3.19.1, which I have not tested.
In general, if you don't really need the latest version of a package, prefer installing the one offered at the official repository. It's much more likely you won't have problems afterwards.