It's not as hard as I thought, but harder than it should be.
In case you don't know, the client software is bundled with the Database Server software - obvious, isn't it?
Anyway, this is how to get it (and you don't have to download everything).
Go to support.oracle.com
Make sure your browser allows pop-ups (thanks Oracle, a throwback to the 90s).
Click on the "Patches & Updates" tab, then on the "Latest Patchsets" link.
If nothing happens, check your pop-ups setting.
If it's OK, a new window will open. Click on "Oracle Database", then use the very clunky navigation method to select the version you want and click on it:
Another window will open, this time with a link to the patchset. Click on that:
The client is hidden in one of these files - to find out which one, click on the "View Readme" button, then on the "software availability" link.
It should take you to this:
So that's the file you need to download for the client.
Why they don't just have a separate link to it I don't know.
