Problem with Google services connection

Hi,
my phone is:

• Redmi 3 Pro
• Android 5.1.1 LMY47V
• Miui Global stable 7.3.2.0.0 (LHPCNDD)

After two weeks of perfect usage, a problem with Google services connection appeared:
can't sync, can't enter gmail, Play Market, can't add new Google account, etc.
I've tried:

• deleting Play Market's cache - no results
• removing current Google account - no success, "doesn't permitted by administrator"
• add new Google account - no success, since there's no connection to Google services.
• reinstall Google services - no success, phone isn't rooted
• back to factory defaults - worked for another two weeks, but then the problem reappeared.

Two weird things:

1. once in a while, in a couple of days, connection reestablishes for a minute and a sync is done
2. after a restart of the phone, connection to Google is good, but disappears after 2-3 minutes.

Is there any things I can try to do before flashing a new ROM?

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