It took some time but Instagram is now letting you save the live videos you take on its service. The reason Instagram finally relented is that perhaps something memorable or exciting happened during your live broadcast and you want to save that for prosperity. 

All you need to do is tap on Save in the upper right corner of the window to save the video. By saving the stream, you won’t be able to save any interaction you’ve had while you were live (i.e. comments, likes, number of viewers, etc.). When you select Done after saving it’ll then be saved in your gallery or camera roll and the live stream won’t appear in the app anymore. The update is coming to version 10.12 of the app on iOS and Android.