Hi, I clicked on "Get it Free" when Mount and Blade was given by GOG a few days ago, but I did not receive the game, not on GOG Galaxy, library, anywhere.

What gives?

There's often (always?) a store checkout process for this, that puts the game in your library. Did you go through the checkout, or just click the banner and leave it at that?

If you click this link it should be the top spot, if its not there then GOG didn't register your click, or if you clicked it then you could have gotten an email about it, check spam and email accounts associated with GOG.com

there was a known SNAFU with mount and blade redemption, just need some proof that you did click the link during the event.

PM sent with a Mount and Blade code.

This. I already owned the game, so it might have been different than for OP, but I got a redeemable code sent via email.

If you don't own it, you don't get a code though. Just directly added to library. At least that's how it worked for me.

EDIT:
Last time I had something like that with Saints Row 2 though. I was fairly sure I clicked it, but few days later there was nothing in my library (got the code from wildhobgoblin anyway). Could have been my mistake though.

In my time here, the vast majority of official, on-site game giveaways/free-with-purchase redemptions have been handled with some sort of (often one-click) "get" button -- usually on the front page, sometimes on some special sale page (like the personal XP tracking pages during a couple of the major seasonal sales last year).
The only giveaway from the last year that I can think of where I had to go to the a game's store page to get the freebie was the first Constructor giveaway this winter. Last one I know of before that was the Worms Forts: Under Siege GA in December 2015, and that one was changed to have a front-page banner "redeem" button within a few hours of its start.

In other words: Other than permanent freebies, very few on-site giveaways in the last few years have required (or allowed) going through the checkout to get the game. (There are getting to be more social-media-related ones that use gift codes that must be redeemed, though.)

ZaineH, thanks man! You are my new best friend.

And thanks to everyone else, although only Zaine's post worked :D

You're welcome! I snagged a spare in case something like this happened.

To be fair, I would have done the same if ZaineH hadn't beat me to it. I hope that makes me at least a friend as well? (yes, I'm the jealous type)