I am fairly certain that neither self-tapping or self-drilling (not the same thing) screws exist for diameters that small. Furthermore, self-tapping threads that are that fine which are required for lengths that short do not seem workable.
You are also probably not equipped to drill it and tap it even if you had the drill and tap. It's just so small and short your that the alignment is going to be off or something is going to cross thread or break.
If you did drill out and re-tap the hole, you would use a 1.25mm drill (which is the tap drill size for a M1.6 screw) and then follow that with a M1.6 tap. And you would not use a power drill, because if you had an appropriate drill and setup you wouldn't have asked your post. You would use a pin vise and do it manually by hand but that makes alignment much more difficult.
Might I suggest pinning it? You would not necessarily need to drill out the hole which would giving you more attempts at it if you mess up. If you can't get a press fit to work, then you cal also flare the pin on both ends of the pin with a punch to get it to stay in.
Also, how did you get that 1.4mm measurement? For the precision you require and the diameters you are working with, caliper measurement should not be trusted.You would need a micrometer for that.
UPDATE:
I did look at on McMaster-Carr and there are M1.4 "self-forming screws for soft metal". Forming a thread does seem like it would work better than cutting a thread for something this small.
So if you trust that M1.4 measurement, then get M1.4, M1.6 and M2.0. I would try the M1.4 first to see if you can't just reform the thread. But it will probably not be removable safely after that. If that fails, then eyeball the M1.6 and M2.0 for which one you think will work. If that fails then you can go with the pinning approach previously mentioned. If you go to M1.6 or M2.0, you would not be drilling it out because you're probably going to mess that up, even if you use a pin vise and to it manually. You're going to be trying to force it in so it will take more effort than normal, but it might also not work and break or strip so you need to judge when it is not working and back off to go with a different method.