The good part is I can continue to play, assuming I do this procedure :

Load my game.
It's raining ...
I wait for 5 minutes.
It's not raining now.
I can play.

@auridulis I use manual saves. I have even used a saved folder I have copied on my disk with a few saves done after the myth in the moutain, and the rain bug is here. As if the game iself was altered for reason.

Though, the saves done before this side quest are not affected. So it may be this side quest fault after all.

@auridulis The fact is : when I loaded my save this morning, there were no rain. When I loaded the SAME save this afternoon, there were rain.
And rain never disapears. I have played one whole hour, it's still raining.

I have completed the "Myth at the mountain" December 24. Since then, I have played the game every day without the rain bug. The whole bug thing went to happen today, 1PM when I was in Canterbury. And it occures in every saves done before.

Edit : it is true that sometimes, if I wait for ~10 minutes, the rain stops. But restarts if I reload my save.

I did the "Myth in the Mountains" mystery a long time ago and played well without rain.
I advanced on the story to the main quest in the Canterbury cathedral.
When suddenly, it started raining .. endlessly.
It started raining right when I entered the Canterbury main gate.
And it rains in every saves I did before.

UBI, you can check my cloud saves, it rains in every of them.

Here is a screenshot : a proof that the raining bug didn't occured right after the "myth in the mountains" : Bassim in front of Rouecister castle in Cent and in clear weather.