Where to fix my iPhone 6 battery?

Click the link and open Bring in for repair and you'll see if there is an Apple Store near you to get your battery replaced --> https://getsupport.apple.com/?caller=sfaq&SG=SG003&category_id=SC0999&symptom_id=99911

https://www.apple.com/retail/marketstreet/

There are also several Apple Authorized Service Providers in your area that will install batteries.

The only caveat is that Apple Stores and AASPs don't guarantee the data. They always reserve the right to do all sorts of things that may mean full replacement or resetting the device. I haven't heard of it happening with anyone I knew for a battery service, but it's always a possibility. If you choose to go with Apple, I'd at least note that the data is more important than anything else and that you wish to get it back if the only way they handle it is to erase or replace it.

Legal - Retail Store Service - Apple

Apple understands that your data may be valuable to you. Data loss during service is always a possibility, and in some cases, data may be unrecoverable, erased, or reformatted during service. For this reason, it is your sole responsibility to back up all existing data, software, and/or programs from your product, and to decide whether to erase any such data from your product, prior to receiving service. Apple is not responsible for loss, recovery, or compromise of data, software or programs, or loss of use of your product or other equipment arising out of the services provided by Apple. You represent that your product does not contain any illegal files or data.