One problem with live and p2p is latency.
What does that mean?
The steps to live are pretty simple on the server side (assuming HLS):
1. Stream to your encoder, ideally at a bitrate higher than the transcoded bitrate.
2. Encode and transcode your video, ideally to 540/720/1080p 30fps. Each resolution will have its own bitrate, so maybe 2/3.5/5.5 respectively. Assume 2 second segments, and a manifest duration of 10 seconds. So you have 5 segments out there at any given time (though there are usually a few more hanging around).
3. Put the 3 newest segments to storage, and rewrite the four manifests with the new segment URLs. (do you need to rewrite the top-level manifest? I believe you do, but I can't remember).
4. Delete the older segment(s) (optional)
So when the client requests the manifest (the m3u8), it'll getch the three sub-manifests (forgot the term) and chose the appropriate resolution. It'll also start loading the segments up. Ideally it would look at the manifest and fetch the latest segment, so it starts nearer to "now."
Then the client will occasionally re-fetch the manifests to get the new segments (the manifest is marked as live; VoD manifests don't require reload). The fetch time probably must be All that takes time. It takes time for the server to encode, time for the encoder to put the file(s), time for the client to fetch the manifests, and time for the client to fetch a video segment.
Looking at the above sequence, a client can be generally 0-10 seconds behind everyone else, depending on how the client behaves. And that's a few seconds behind "live," because receiving, encoding and putting files takes time.
So can you do p2p live? As long as you relax the constraints on what you mean by "live," yes. As you can imagine, the chain of latenty keeps growing longer the more peers a segment goes through. And that segment is only really good for 2 seconds (or up to 10 seconds, if the client is sloppy). If live means "up to 20 seconds since now" then yes, you can definitely do it. The tighter that time window gets the less likely you'll be able to do it. You might be able to do it with a lower bandwidth stream, but even TLS negotiation takes time. Does your client not use TLS? That will save you time.