What you need is a two fluted high helix micrograin carbide end mill. Here in Canada two makes are Garr and Gorilla but there are many others. The high helix which acts to pull the chip up out of the cut, micrograin carbide can be sharpened to a very keen cutting edge and two flutes give good chip clearance.
You don't mention whether you are using coolant or lubricant, certainly you should be. Flooding with collant could be messy but a mister could spray a small amount of coolant on the tool and this will help immensely.
With a 6mm cutter running at 6000 rpm in 6061 aluminum alloy with coolant you should be able to cut at least one diameter deep at the feed you are using.
An open mind is a virtue...so long as all the common sense has not leaked out.