Same issue here on my iPhone 6 plus. Been hugely frustrating since I originally upgraded to iOS 9, and has continued with every update since then including 9.1. There are many press articles about these problems with iOS 9, and Apple has admitted these problems exist. But they said 9.1 would fix it and it didn't.
The problem is, these problems don't occur on all or even a majority of iOS devices. Thus, it was easy for Apple to hide this one under the rug until enterprises who use iPhones and iPads for business started complaining about it. I work in this industry and advise many such organizations on mobile computing strategy and issues, most involving iOS. I can tell you that these problems are having a business impact (in some cases, major ones - as when iPhones/iPads are used as point-of-sale devices in retail businesses.) In working with these businesses (most I work with are Fortune 1000), I'd estimate about 20% of the devices out there might be affected. It smells like an issue with particular firmware versions (which can vary amongst devices of the same model) or maybe even processor manufacturers (there are two processor manufacturers supplying iPhone 6/6+ devices, and any given iPhone 6/6+ might have one or the other.) Firmware version issues are the most likely, though, given the estimated percentages of affected devices I cited above.
If Apple can't get this stuff fixed and soon, they will have some organizations re-thinking their mobile platforms moving forward. Android-for-Work is waiting in the wings, and it's a viable enterprise platform. I'm already having discussions with CIOs seriously considering it, given the impact these 9.x problems have had on their businesses.
In the meantime, many, like myself, have essentially a $900 brick. I have to soft-reset the phone 10 or 12 times a day due to a non-responsive touch screen. Four point-releases later, nothing has changed. It's extremely disappointing, and I hope this isn't a sign of things to come from Apple in the future.
I can tell you for a fact that this is not isolated, not "stupid user issues", damaged hardware problems or the like. It's a serious spectrum of related issues that aren't "fixable" by the end user, other than factory-resetting the phone to get back to a stable iOS 8.x version. The majority of iOS users are not affected, but a very significant population of unfortunates are, including some of the biggest enterprises in the world.
BT