I think the story of Sneferu in the Westcar Papyrus might also encodes Pi as 22/7 but in mythology. The story starts on page 65 here: [mjn.host.cs.st-andrews.ac.uk]
In essence, I believe there are 22 people on the boat - the 20 female rowers + Sneferu + Djadjaemankh = 22 people
The story tells us that one of the girls rowing snags the oar in her hair and her "fish-pendant of new turquoise fell into the water." page 88. Following the story through, it tells us that she was the stroke rower for her side and her side stops rowing. It doesn't say the other side stops also, so I presume they continue to row. In these circumstances the boat is making a circular motion.
The fish pendant of new-turquoise is at the bottom of the lake. The first phonetic character in the fish pendant is a zig-zag line with 7 peaks. So we have 22 people on the boat which is on top of the fish pendant with 7 peaks in its first phoneme.