I'm experiencing this as well, although I hadn't considered the impact of cold weather until today. I am using an iPhone 6 (not 6s, not plus) that I've had for about 18 months. My wife and daughter have the same exact phone, though, and are not experiencing this issue at all.
With my phone, I've noticed during colder (below 30 degrees F) weather that the reported charge on the phone will very quickly drop off from 100% to the 60% range then to the 30% range and then cut off altogether with little or no warning. Last night, in particular, I had the phone plugged in and charging on my night stand. Before I went to sleep, I checked it and saw that it had charged to 90%, so I went to sleep expecting to find it at 100% the following morning. I awoke at around 3am to go to the restroom and checked the phone on my way back to bed, just to see what time it was. Instead, I found the empty battery with a red line. I was not expecting to find this. Sometime later in the morning, the phone came on and played the "awakening" sound and illuminated the ceiling in my bedroom with bright white light. At that point, the battery was reported to be 36% charged. It has remained at 36% charged throughout the day, in spite of nearly constant usage on my part.
There appears to be a recall on batteries for the iPhone 6s, but not the iPhone 6. I checked my phone's serial number and it is not covered by this recall. I feel that it should be. This is not proper behavior for a lithium ion battery in any weather.
For now, I've made an appointment to go to the Apple Store tomorrow and have them look at it. I suspect it'll need to be replaced, so I'll be out $79 + tax for that.