If it's a one off thread the end of a bit of silver steel (outside diameter same as one of your collets) and drill a 1mm hole in the other end and loctite a 1mm bit in (when, rather than if, you break it warm with a blow lamp and get the broken bit out…)
There are stacks of ER11 collet holders on ebay and the like, put one of those in a collet and tighten it up… Then you can hold any size of small drill or cutter The odds of it getting pulled out by a 1mm drill are small 
The truth is that you will struggle to do 1mm holes with your machine, it doesn't go fast enough and raising the knee will require a lot of subtlety not to break the drill… I too have a heavy weight machine and it's hard work with small drills (at least they are cheap!)