Since Instagram launched the story feature, lots of updates and improvements have been added to increase the user engagement. Not long ago, Instagram added the live video feature to insta story and now they have just added another new feature.

Instagram has just added an option to save live video. The new feature which was added to the V10.12 update is available for both Android and iOS users. To use the new feature, just click on the save option at the top right corner of the screen at the end of an Instagram broadcast.
For now, you can only save the live videos. Other things like comments, likes, number of viewers or other live interactions will not be saved. To view the saved live video, just go to your gallery or camera roll after tapping done.

As I said earlier, this option is already available on app store and Android play store. Just head over and update your app to the latest version 10.12.