Does the NEC require armored cable in kitchen islands?
The relevant NEC is 334.15B


  (B) Protection from Physical Damage. Cable shall be protected from physical damage where necessary by rigid metal conduit, intermediate metal conduit, electrical metallic tubing, Schedule 80 PVC conduit, Type RTRC marked with the suffic -XW, or other approved means


This is more of a "It's a really good idea if you do this" area (with regards to the island). It depends a LOT on the design of the island itself. I've seen BX wires that were overkill and I've seen NM wires that were unsafe the way they were routed.

What I would do is find an electrician who isn't in the Always BX camp. Look at the design with them and figure out if there's anything that could cause damage to the wire. If it's always going to be in an inside corner, the risk is low enough you could get away with NM (and I would say most islands could). If the wire will have to run tight to, say, metal tracks for a drawer (where it could make contact), then run BX to be safe.