what are you making that you need wood only 6mm thickness ?
That is quite thin and a big ask , I have made work in different woods but that thin it wood be birch ply the popular wood is quality birch plywood and a lot of plans are for 6 ,12mm ,18mm etc
at the moment I am using 20mm pine , Geoff on here makes his stuff from I think 18mm hardwoods , if it needs to be 6mm hardwood then you are going to be limited buying from stock I guess you could use a thicknesser or bandsaw to get that thin
mark