I had this same problem (the periodic dropping of the web connection with the WAN disconnect messages in the log) with the 150916 firmware, but it went away when I upgraded to the 160517 firmware (for the C9v1) available on the ".com" site (but not the .us site). This latest firmware has been rock stable for me. I know it's a few months since anyone has posted in this thread, but I figured I'd let people know in case they stumble upon it.

I upgraded to the newest firmware available to me when I got the router ( Archer C9(UN)_V1_160517 ). After a while I noticed this problem where the WAN light turns orange and I lose connection for 20 seconds. It comes back on and all is fine, but this happens several times in an evening. I saw some people mentioning it happened more at night. It happens all the time for me. Some people said downgrading to Archer C9_V1_150507 fixed all of their problems.. I downgraded to this and had the same problem except the downtime is less frequent, but it lasts for 1-2 minutes. Any suggestions here? I like everything about this router, but this issue is impossible to look passed.

Have you checked your isp for troubles on there end
I know Virginmedia in UK are flooded with speed issues and a silly update to the superhub 2ac which I have and my archer c9 v3/ v4 is making my c9 play up to the point I willing to change to bt UK