If you've got a 6s, you might be eligible for a replacement:

https://www.apple.com/uk/support/iphone6s-unexpectedshutdown...

I'm pretty sure my phone is affected, but according to that page:

    To prepare your iPhone 6s for the battery replacement process, please follow the steps below:

    Back up your data to iTunes or iCloud.
    Turn off Find my iPhone
    Erase data and settings in Settings > General > Reset > Erase all Content and Settings

Just to replace a battery??

They swap out the entire phone, and send the old one out to be refurbed and sold.

Not anymore, at least the NYC stores don't. They take your phone in and give it back to you in a couple of hours(depending on how busy they are). That's how they did it when I had mine replaced last week. The 5th Ave store didn't have any batteries in stock so I had to either wait a few days or go to a different store that had them.

You should never send a phone with your personal data out in the wild. If an Apple Genius is going to take your phone to the back of the Apple store and out of your sight, back it up and restore it to factory settings.

They might be sending you a refurbished phone instead of replacing the one in your existing phone

It's always prudent to backup your data if your device leaves your possession. What if FedEx loses the package or it's stolen off the truck?

Well, in all fairness with regards to cold, this is going to happen with batteries - it's just their nature. When it's -20ºC out, my old iPhone 4s can only be out for so long before the already depleted battery just dies off.

But this is just the nature of batteries.

I've had a 4, 5, 6, and 7 and I've never had this happen once. Your comment seems more like a "LOOK IPHONE DOES IT TOO" than anything meaningful.

My 6 has started doing this just after its 2 year anniversary. I bet it has something to do with battery age as well

This can be "fixed" by resetting the battery data on the phone and then letting it completely discharge before charging. I've had it work myself several times (I own two of every model iOS device for app testing purposes).

Exactly the same for my 2-year old 6.

My 5s does it if I don't recharge it fully.

iPhone 6s seems to have this particular problem

If you have an iPhone 6S with this problem you can get it replaced: https://www.apple.com/support/iphone6s-unexpectedshutdown/

Yes. I deal with this.

It's called an anecdote. I think that's useful in this situation.

My iphone 5 does it. Tends to be around the 20% mark.

Yep, my 6S had a goosed battery, walked into the Apple Store and 15min later I had a new phone with a predictable battery (and a headphone jack, happy times)