1

あんなにコチコチな座り方をする猫なんていやしませんぞ。
A cat that sits so rigidly ???

I can't figure out what this いやしません is.

Since the negative form of いやす seems like a ridiculous proposition in the context I can only assume that it is いや + する, and that いや is possibly something to do with being disagreeable, but I really can't get it to make sense.

What is this いやしません? Is it common? Any more examples?

2

I think I might know what it means, but I apologize if my explanation is not that good. I believe the verb here is 居る (iru), thus, your sentence would translate like "There's no way that there is a cat (or cats) that sit in such (stiff) manner."

いる is not written a lot of times with its kanji so maybe that's why it was not so clear at first. In the sentence, there is the verb masu stem (い from いる) +やしない.

This is related to the ~はしない/~やしない grammar which you can read about here if you would like.

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