Yes, broken promises have been the basis of perjury convictions.
Perjury statutes vary from state to state, and so does the answer to your question. But the way most perjury statutes are structured, there is no reason that a broken promise could not be the basis of a perjury conviction.
Oaths are often grouped into two classes: (1) assertory oaths, which are oaths about the truth of the facts a person is relaying; and (2) promissory oaths, which are oaths about how we intend to act in the future.
The oath taken before testifying in a deposition or trial is usually thought of as an assertory oath. Promissory oaths are the kind you hear the president take when being sworn into office, or that immigrants take during naturalization ceremonies.
False statements about what happened in the past, made subject to an assertory oath, are the classic case of perjury, but perjury charges could likewise be sustained for false statements about what one intends to do in the future. For instance, in Norton v. State, 5 Ga. App. 586 (1909), the Georgia Supreme Court upheld a conviction for false swearing against an election official who swore he would "fairly, honestly and impartially" administer a primary election but then altered the results:
Our statute against false swearing was intended to punish the making of false oaths in other than judicial proceedings, being in this respect alone distinguished from the statute against perjury. Since perjury is limited to judicial investigations, and since these investigations generally relate to things which have already occurred, it is not remarkable that cases have not arisen involving the question as to whether perjury could consist in the violation of a promissory oath. The subject-matter of false swearing is not so limited, and may relate to the future. Morally speaking, it is as culpable for a person to swear that he will not do a thing, and then knowingly and wilfully do that thing, as it is for him to swear falsely as to what he has already done.
However, this seems to be the minority position. See, e.g., State v. McCarthy, 255 Wis. 234 (1949) ("Violation of an oath of office does not constitute perjury as that offense is defined in the law."); People ex rel. Bryant v. Zimmerman, 241 N.Y. 405 (1926) ("[Promissory oaths] may not be the basis of a criminal prosecution for perjury."); State v. Vause, 84 Ohio St. 207 (1911) ("The authorities agree that no officer can be punished for perjury by reason of nonfailure to comply with the conditions of such oath.").
In the assertory-oath context, it seems that perjury charges based on statements of intent to act in the future are fairly uncommon, but they do exist. Statements about a person's residence or domicile provide a good example, as the legal definition of these terms goes beyond the common definition to mean not just the place where a person currently lives, but the place where a person intends to return whenever they are away.
For various reasons, it can be financially beneficial for a person to spend extended periods of time in one place without other parties knowing that they either have no intent to stay or no intent to leave. Perhaps you want to claim a propety tax break for property that you don't actually live in, perhaps you want a better mortgage rate available only to first-time homebuyers, or perhaps you want to avoid paying sales taxes on an online transaction.
In any case, claiming residence in the favorable jurisdiction necessarily involves representing that you intend to return to that residence in the future. When you make that representation under oath, you are at risk of a perjury charge.
One example is Ind. Code § 35-44-2-1(a)(1), which makes it a crime to make "a false, material statement under oath or affirmation, knowing the statement to be false or not believing it to be true." In White v. State, 25 N.E.3d 107, (Ind. App. 2014), an elected official councilman moved out of the district he was elected to represent, and then submitted a variety of statements under oath stating that his residence was actually his ex-wife's house, which was in his old district.
A jury convicted him of perjury for lying about his residence on his application for a new marriage license and on his voter registration form. On appeal, the court reversed the perjury conviction based on his marriage license application, because his precise street address was not material to the application, but it affirmed his perjury conviction based on his voter registration.
While it seems relatively uncommon to premise a perjury charge on a false promise made under oath, White points out the additional hazards associated with such a false statement. Besides the perjury charge, he was also convicted of mortgage fraud, voter fraud, and theft based on his misrepresentation about whether he intended to return to his old address.
And there are plenty of other examples of criminal convictions based on false promises. For example, a German immigrant's false promise, under oath, to forswear allegiance to the Nazis resulted in his being stripped of his citizenship in Knauer v. United States, 328 U.S. 654, (1946).