I can't make game saves in my GOG versions of Omerta, Pillars of Eternity or Divinity: Original Sin Enchanted Edition.
Also current installation of Dungeon Rats doesn't work; Age of Decadence which is technically similar does, though.
These troubles doesn't affect all of the games, for example in Wasteland 2: Director's Cut saving works fine.
I haven't tested other games in my GOG library yet.

I'm suspecting that it has something to do with free Avast anti-virus, but I would like to hear if anyone else is having these sorts of problems.

And is there any free anti virus software that anyone can recommend.

Have you tried running the games as Admin?

I've been using Comodo for years, has kept me safe so far.

Ok. Run as an admin seems to do the trick.

Tip, never ever install any game or application in the windows folders. Create something like c:/mygames on your machine and install games to that, avoids the wi does permissions issues like this.

Well... some games like to make the save files in the directory where they are installed, while others use special locations like %appdata% or just My Documents folder.

I suspect that since you said that running as admin seems to do the trick that you are probably running the games from Program Files folder or another location that Windows has some kind of write protection on.

To avoid permission issues, as a general rule, it is suggested to install the GOG client and games either to the root of the system partition (Ex: C:\GOG, C:\Games) or to a different partition.

Oh come on people, those partition things are basics.
Besides I've had some of those games installed for over year on my hd with no issues what-so-ever.

New version of the Avast really seem to be what causes these issues.

hmmm.
2 broken game installs (Dungeon Rats and Serpent in the Staglands), which got fixed with simple re-install.
But it is Divinity: Original Sin that troubles me. Manual saving doesn't seem to work, but autosaves do for some reason.
I need to dig more info before I dare to try to launch that game again.
Pity; I really wanted to play it during this weekend.

Also. pfeffff. Checking 150+ installed games and working of game saves in those takes some time and willpower.