For 1A you can go stupidly small; but for 120 or 240 supplies it is matter of adequate insulation and more importantly how it is protected, rather than physically smallest wire.
For safety regulations there must be a fuse beforehand that will blow without the wiring heating dangerously in case of a fault; that fuse defines what wire you can use.
If you have a 1A fuse on the supply connection then you could easily go down to 24AWG (0.2mm^2), that's rated 3.5A
We use 0.5mm^2 routinely as both single core and YY multicore in industrial machines, for low current circuits at up to 240V. That has a nominal rating of 9A in open air.