After introducing “Stories”, Instagram added some cool features such as “Live video”, “Boomerang” and “Hands free” to make it unique and different from Snapchat. However, the live video feature had some limitation and the user had no option to save the Live video like Facebook. Recently, Instagram has enabled this feature, where you can get a save icon on the top right corner of the phone screen just after finishing the live broadcast and by tapping on the save button you can save the complete Live video to your phone. Make sure to tap on Save before tapping on the Done button.
This feature is available for both Android and iOS. Worldwide InstaMeet 15 (#wwim15) will be celebrated on 25th and 26th of March 2017. If you wish to do live broadcast at the InstaMeet, update your Instagram account before going to the event at your city so that you can also save them for later.
— Instagram (@instagram) March 20, 2017
Source: Instagram blog