Thank you for your responses.
I opened up the panel under my shower tub (yes, it is a rear bath). Here is a picture of what I found:
The pipes are dry and they don't look like they are the culprit. I checked the valves at the end of those pipes and they look fine as well.
If I thought it possible, I would say that the water is seeping through the floor. Is it possible?
It is possible that the damage was caused by frozen pipes. Although my dad had the problem a few years back and had it fixed.
After thinking about it, I am more scared by the rivets than by the belly pan. I have no idea how to drill them out and how to replace them. I know I will have to know soon enough so now is a good time. Yuk!
I am thinking that the leak could also come from the toilet. I will change the donut this week and see if I find anything interesting.
Following your instructions, I will find all the pipes and draw myself a diagram. I do have a copy of the owner manual but it is not helpful. I will probably have to buy a copy of the service manual from Airstream.
I will also buy myself a water pressure regulator. I will inspect the water pressure relief valve (if I find it).
I did not check if it leaks when using the water pump. I don't even know how to make it work. I assume that I would have to fill my water tanks and then unplugged the city water. Would it start on its own once I turn a tap on? Is there a switch I have to turn on before?
Thank you for your answers. I will keep you posted.