What exactly makes it part of the body? The attachment itself?
And if the arm is connected simply by some special holder, and the control is done entirely by a chip that transmits waves that create the communication — meaning there is no physical connection of electrical wires to the body / brain / chip.

What would be the difference between that and a wheelchair that has some sort of tube-like connection from the head to its operating system.

I'm talking a bit like science fiction.
On the other hand, the Sages talked about a flying box, so I think I'm in good company.

This isn't science fiction. Halakhic decisors have discussed cochlear implants in the ear at length, and even glasses (regarding carrying on the Sabbath), which are seen as part of the body. It isn't connected to the question of how the connection is made, but rather to the functional role.

It sounds reasonable to go by the rules of carrying.

But is there not a difference between a cochlear implant and glasses, whose use is passive (the implant is activated before the Sabbath), versus an arm, whose function is active?

And maybe it's problematic in this case to go by the rules of carrying. For example, an electric / digital watch — you're allowed to walk around with it as long as you don't use it beyond its passive function.

And another question along the same lines: why, really, prohibit an electric wheelchair, whose whole role is basically to replace the legs? (I'm not talking about the modified electric vehicle of Tzomet.)

I didn't say to go by the rules of carrying. I brought them as an example of this kind of consideration. I didn't mean to claim that everything would be determined by the rules of carrying, since I haven't examined that thesis, though on its face it seems fairly reasonable.
Whether the function is active or passive really makes no difference. Our natural hand also has an active function, and we are allowed to send it electrical commands from the brain that activate it. An electric wheelchair is similar, but not entirely. It doesn't function as a limb, but replaces it with something else, and that something else is also operated like a device, with switches and pressing buttons, and not from our brain. That may perhaps be worth discussing.