Quick answer: The microphone directive controls whether a document or embedded frame may access audio input devices through MediaDevices.getUserMedia(). Disallow it everywhere with the response header Permissions-Policy: microphone=(). Allow only your own origin with Permissions-Policy: microphone=(self). When blocked, getUserMedia rejects with a NotAllowedError and no microphone prompt appears.
user-consent

microphone

The microphone policy controls access to audio input devices requested through the NavigatorUserMedia interface.

If disallowed in a document, then calls to getUserMedia() MUST NOT grant access to audio input devices in that document.

How to apply this policy

Send the following HTTP header to control the microphone policy, and disallow it on all origins:

Feature-Policy: microphone 'none'

Does it work?

Currently Firefox, and Chromium based browsers, such as Google Chrome, Samsung Internet, and Opera, are the only user-agents to support Feature Policy. The minimum version that correctly recognises the microphone policy is:

Mozilla Firefox

65+

Google Chrome

60+

Microsoft Edge

Not supported

Apple Safari

Not supported

Discussion

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