Majority of the search results for "download periscope" lead to websites allowing me to download the app.
"download periscope stream" is a much better search term that actually led me to two solutions but none of them are convenient.
1) tried and works
http://www.quora.com/How-do-I-save-my-Periscope-video-or-broadcast-to-my-phone/answer/Andrew-Leyden - that requires that the phone is connected all the time
(for the whole 50 minutes of the stream)
2) didn't try, should work
Recording screen + using http://www.instructables.com/id/How-to-record-audio-from-your-computer-using-Quick/ to record audio
(again, 50 minutes waiting)
I also tried using technique described here - https://stackoverflow.com/questions/30073330/downloading-storing-periscope-live-streaming-broadcasts - just by observing network request and populating list of URLs to download via wget:
https://replay.periscope.tv/ ... ==/chunk_1.ts
https://replay.periscope.tv/ ... ==/chunk_2.ts
But that naive version fails - they check cookie... Maybe I should use PhantomJS to accept cookie and then download the chunks? Before I start digging into that - maybe there is a straightforward, off the bat, ready to go solution I'm missing?
The question is generic and applies to any periscope video.
Bonus question - what if the video is mine - can I retroactively save it to the camera roll from the periscope app? (assuming "autosave broadcast" option was turned off)