5C won't keep time

My girlfriend has an iPhone 5C on Verizon with the latest iOS version. She noticed a month back that is stopped keeping time. We'd set the time and within an hour, it would be off, like it never moved forward.

We tried resetting and restoring the iPhone in iTunes with no avail. The issue persists whether we manually set the time or leave it to auto-set from the carrier.

Brought it in to the Apple store and they claimed they have seen it before and it's an hardware issue with the "clock" -- which I'm not convinced since there is no "clock" in phones -- it's all software based. Irregardless, if they've seen it before, it seems like it's a defect. Since it's 14 months old, it's just out of warranty, so they wouldn't swap it.

Not sure what else to do or who to contact.