Latest known working firmware 1.0.9.48 (Mi Band version 1.3.312). Sadly Xiaomi have made reverse engineering the band protocol very difficult with obsfucation and the removal of debug logging within the app.
The app currently only works with Lollipop due to significant incompatible changes in the Bluetooth stack, there are no immediate plans to change this.
Disclaimer: This app includes no warranty for your device, in my testing I've never bricked my band - but as I'm reverse engineering their API, it's entirely possible, albeit unlikely.