Update 5 Jun. 2023 - General Availability (GA) release
Microsoft released Version 1.0 (General Availability (GA) release) of the SQL Server Integration Services extension for Visual Studio 2022.
Update 24 Nov. 2022 - Public preview released
Microsoft added a public preview of the SQL Server Integration Services Projects 2022 to the Visual Studio marketplace, which is tested against Visual Studio 2022 17.4
Read the announcement
Update 19 Sep. 2022
Microsoft Data Platform MVP Andy Leonard stated that "I communicated with the Microsoft SSIS Team. They shared they are targeting a preview of VS2022 Integration Services Extension for the end of CY22".
Update 12 Jul. 2022
Microsoft announced the support for Visual Studio 2022 via the Microsoft Analysis Services Projects 2022 extension.
Update 01 Jul. 2022
Microsoft mentioned that Visual Studio 2022 support is one of the major items they are working on, and it will be released before the end of the year.
Update 23 Jun. 2022
Microsoft released SSIS 4.0, adding support to SQL Server 2022, but still, this version does not support Visual Studio 2022.
In addition, the support of Visual Studio 2022 is added as a common issue
Update 23 Feb. 2022
Microsoft released SSIS 3.16, where the support of Visual Studio 2022 is removed from the SSIS Installer (Bug fixed).
Initial Answer - 16 Feb. 2022
Microsoft SQL Server business intelligence extensions are still unavailable for Visual Studio 2022.
Many people are still confused about the difference between SQL Server Data Tools used to create SQL databases projects and the business intelligence suite:
Using Visual Studio 2022, only the SQL Server Data Tools required to create database projects are currently available. As mentioned in the official documentation:
The core SSDT functionality to create database projects has remained integral to Visual Studio. The extensions for Analysis Services, Integration Services, and Reporting Services projects are currently unavailable for Visual Studio 2022.
You can note that in the Visual Studio marketplace, only Visual Studio 2019 is mentioned: