There might also be a proof from China, which might even predate Pythagoras. It can be found in the Zhoubi Suanjing, a book dated to c. 100 BC, but which might contain earlier material and is attributed to Shang Gao, the astronomer of the influential Duke of Zhou, who lived in the 11th century BC.
Finding out what the deal was, I had to dive deep into the literature and I even contacted a historian who had worked on the translation of the document. It finaly led me to update the wiki page on the topic.
These were my conclusions. The book definitely makes use of the Pythagorean Theorem on various occasions and might also contain a geometric proof of the theorem for the case of the 3-4-5 triangle (but the procedure works for a general right triangle as well). But unfortunately, the text isn't very clearly written and only a favorable interpretation, with commentary by a modern scholar makes it possible to read a proof in the text (See Geometrical Figures and Generality in Ancient China and Beyond by Karine Chemla). Given this interpretation, however, it does give an elegant proof of the theorem!
In the 3rd century AD, Zhao Shuang (3rd century CE) did add a commentary to the text, which included the diagram depicted in this post, which might correspond to the geometric figure alluded to in the original text (although Zhao only refers to the image in another context). The image does make the interpretation of a proof more plausible, as this figure can indeed be used to proof the theorem.
"The Great World History Book"