WhatsApp has started rolling out View Once voice messages globally, and it should reach all users in the coming days. Similar to View Once photos and videos, voice messages will also be marked with the "one-time" icon, and can be played only one time.
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This feature works only on mobile devices for added privacy as you can't open View Once photos and desktop on WhatsApp's desktop clients.
WhatsApp has been testing this feature on its Android and iOS beta apps. It's available to select beta testers who were able to try this feature out before the official release. Since the rollout has just started it might take some time before the feature reaches everyone. You can still check by following the steps mentioned above.
This feature does seem safe and private but do note that on Android devices it's possible to record the View Once voice messages with the native screen recorder. So it's advisable to still be cautious while using this feature.