It's been 3 decades since it was shown that there is no projective plane of order 10. I can find lots of commentary and references on the state of play on the attempt to show there is no projective plane of order 12. It seems to be an open question whether there is a non-desarguesian plane of order 11. I can't find any research on that, however. Any information would be helpful.
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@user10354238 yeah, I know that, that's the unique desarguesian plane of order $p^m$. Please reread the question. I am asking about non-desarguesian planes, of which there are none of order less than 9, a complete list (of 3 of them I believe) for order 9, and order 11 is an open question. – C Monsour Apr 02 '21 at 12:47
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I have not seen much work toward this (I would imagine it is a very difficult problem; it might be easier to disprove the existence of an order 12 plane than to decide the existence of a non-desarguesian order 11). – xxxxxxxxx Apr 02 '21 at 15:05
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However, I did find a 2010 PhD thesis of Cafer Çalışkan that looks at trying to construct an example by looking at subplanes of the known nondesarguesian planes of order 121. – xxxxxxxxx Apr 02 '21 at 15:07
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1(It is 100% an open question, if that is what you are asking) – xxxxxxxxx Apr 02 '21 at 15:07