Just updated SSMS 2016 using the July update release. The install itself went smoothly. However, it won't connect to any databases anymore... I'm apparently not the only one with this problem, as I see recent internet postings by other people who face exactly the same problem:
Cannot connect to PC\SQLEXPRESS.
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ADDITIONAL INFORMATION:
Method not found: 'Void Microsoft.SqlServer.Management.Common.SqlConnectionInfo.set_ApplicationIntent(System.String)'. (ConnectionDlg)