Smokiness is generally derived from peat, and the two are certainly related, but they are not always found together. For instance, I perceive the peat in laphroaig 18 to be phenolic, medicinal, and briny, not really smoky. Some folks perceive other flavors that are not caused by peat influence to be "smoky." I notice it occasionally in sherry and port finished whisky that is mostly or entirely unpeated, but it's not really smoke in the same sense as Ardbeg smoke. Additionally, not all peat and peat-smoked malted barley is created equal. Port Ellen Maltings on Islay provides the vast majority of barley for Islay distilleries, and they generally seem to work with clients to provide them with malted barley peated to a wide range of levels. My understanding is that the malted barley that becomes Caol Ila and Lagavulin is all the same. My understanding is that Ardbeg, laphroaig, Kilchoman, Bruichladdich/Port Charlotte, and even Connemara (which I personally hate) use at least some Port Ellen Maltings barley, though they all specify barley that has been smoked to different ppm. Springbank distillery does their own malting, and they appear to use peat from Campbelltown, which provides very different characteristics to the spirit vs. Islay peat. I don't get a lot of "smoke" flavor from Springbank, but there is no mistaking the peat phenolic peat influence. Longrow is peated to about the same level as Laphroaig (around 55ppm), but again, the campbelltown peat influence is very different. Talisker and Highland Park, two other non-Islay generally available peated whiskies, aren't as far off Islay characteristics as Springbank is, but they are certainly different. I don't know where they get their barley (or the peat that is used to smoke the barley) from though. For all I know, it could all come from Port Ellen, but who knows.
Another interesting tidbit is that most, if not all, of Diageo's distilleries actually age their whisky in a centralized facility owned by Diageo. So the idea that most Islay whisky picks up brine and other characteristics generally associated with peat flavors as they age on Islay is either not true or Diageo is doing some interesting stuff to the casks as they age.
Here's a Ralfy video on the subject of peat: