Plumbing Questions? Ask a Plumber and Get Answers ASAP.
Is the handle attached to a short stem coming vertically out of a dome or cone shaped escution? Or..... is the handle flush against this dome or cone shaped piece? Model Number of faucet if available.
Holding on to the handle with one hand, attempt to unscrew the escutcheon itself counterclockwise. let me know if that gets it.
Give me a few minutes to pull it up online. I'll send you a link then pick the number that it appears to be.
http://www.peerlessfaucet.com/products/function.do?function.areaName=Kitchen&function.brandName=Peerless&function.functionName=Two+HandleAfter looking through the selections for two handle. I am going to switch gears here, sorry. The little hole should have a very small allen set screw. It may just be corroded with water deposites making it difficult to unthread or even get the allen to set in there properly. Try spaying a small amount of Liquid wrench (WD-40) and see if that will help. Careful not to let it remain on the finish long as it can ruin that. just wipe it off well.
Hopefully this will help you. It looks like an allen set screw for your handle to me according to these instructions.