I had the same issue today. I have iPhone 6 and had it attached to my bike on a bike trip today. It was about 10° C outside (50° F). I had about 60% battery and it turned off without any notice. When I tried to turn it on, first it said I should plug it in to recharge, when I was holding the power button, it started and then turned off after about 10 seconds of me using it. I've tried this about 4 times with some breaks between, but it was the same each time.
Then I really needed to look at the map for directions, so I was trying out various tricks to get it working. I turned the phone on and then very quickly turned on Airplane mode and then went to settings and turned on Low battery mode. After that, the phone didn't turn off and I could use it. I even turned off the Airplane mode after some time and it was still working and it didn't turn off after that. So maybe the Low battery mode helps with this?
After that, it didn't turn off for like 2 hours and was working just fine. When I came back from outside, the phone had about 27% of battery (I was using it a lot for navigation etc.) When I plugged the phone in (still quite cold), after only few seconds of charging, it reported it had 41% of battery. Weird.
My phone is about 1.5 years old, but in a good state generally.
It seems that it has some issues with battery in cold weather and it doesn't even have to be that cold... I wonder if I'll have more issues when the winter starts, but unless cycling, I usually keep the phone in my pocket, so I guess it should be fine.