I had similar problem with Sony Xperia Z3 Compact. After upgrading to Android 6.0.1 Google Play Market was unaccessible. The program just crashed without any notification from the system. After playing around I added permissions to use memory for all applications including Google Play Services. Then, after a few attempts, Google Play Services could be downloaded and installed... I'm not sure but probably the main reason was that some applications data were relocated to SD card before upgrade to Android 6.0.1 and new system introduced some additional rules for accessing SD card, so Google Play Services just could not store something in SD card without additional permissions and crashed.
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