The job of the Speakership is not dependent on being a sitting representative. It can be from within the ranks or outside the ranks.
However, there is no time in US History of which I am aware, in which any state, or even colonial legislature for that matter, elected a non-member of their chamber to be their Speaker. So until it actually happens, it is only a theoretical possibility.
However, I would point out that if the House were able to elect a non-Representative to be their Speaker, then the Speaker would not have a vote in the Chamber since only elected Representatives are constitutionally allowed to be members of the Chamber as indicated by Art. I Sec.2 Par. 1 and 2 of the US Constitution which reads:
"The House of Representatives shall be composed of Members chosen every second Year by the People of the several States, and the Electors in each State shall have the Qualifications requisite for Electors of the most numerous Branch of the State Legislature.
No Person shall be a Representative who shall not have attained to the Age of twenty five Years, and been seven Years a Citizen of the United States, and who shall not, when elected, be an Inhabitant of that State in which he shall be chosen.
“The job of the Speakership is not dependent on being a sitting representative. It can be from within the ranks or outside the ranks.”
No one believes that the Chief Justice of the United States can be someone other than a Justice of the U.S. Supreme Court, and, until a few years ago (when a couple of Republicans upset at Newt Gingrich voted for retired Republicans for Speaker) no one other than a sitting Representative had even received a vote for Speaker. I think the theory of the non-member of the House serving as Speaker is an interesting exercise in constitutional analysis, as is the theory that the Governor of New York could be in the line of succession to the presidency (a governor is, after all, an “officer”), but having a non-member serve as Speaker ultimately would be a distortion of the Framers’ original intent.