The basic procedure is this:
1. Find the maximum load to be served so you can supply it correctly. 210.19 A 1 - In your case the load is very low, so you don't have to worry about that.
2. Find a breaker that can supply the load 240.4 - In your case, you run up against the "normal" minimum breaker size of 15A that is readily available for most breaker panels Table 240.6 A. Although 3A is a standard size (240.6 A) but good luck finding that in a standard breaker. So technically you could install a 3 amp breaker using #18 wire rated for 5.6A (240.4 D 1) BUT...
3. Size the wire for the breaker, considering any derating you might have 310.15 B
4. Make sure there aren't other rules that supersede these like 310.106 A which sets the minimum wire size at #14 copper, unless specified elsewhere. And rules like motor loads mentioned by electrofelon, 430
5. As augie47 mentions, consider supplemental overcurrent protection for your particular device if needed. But this doesn't replace the other requirements. 240.10
So your stuck using #14 wire, but you can use a 3A breaker or fuse if you want (if you can find it for your application)
This is just the bare basics, of course.
(I hope I got all this right...)