The Prophet (peace and blessings of Allah be upon him) restricted obedience only to that which is right and proper.
Nobody, neither his best companion nor his beloved daughter was infallible.

But what about the hadith: “Fatimah is part of me and whoever angers her angers me.” [Bukhari no. 3523 and 3556; Muslim, no. 2449].

(Note, Rafidis often present a similar narration, however, they post weak and fabricated wordings of it and add ‘Bukhari’ as a reference)

First of all, when the Rafidis quote this hadith as evidence for criticizing Abu Bakr (ra), because of their ignorance and trickery, they conceal the fact that criticism would be directed at ‘Ali (ra) first of all as the reason why this hadith was narrated was ‘Ali’s (ra) wish to propose marriage to the daughter of Abu Jahl, as the result of which Fatimah (ra) became angry. Of course, we couldn’t care less if they deny the authentically narrated context of the incident.

Anyways, this issue was resolved over a century ago and Ahl al-Sunnah focus on Tawhid, not on such bygones. We neither speak badly of Abu Bakr (ra) nor of ‘Ali (ra), we are not obsessed.

Besides, the hadith has nothing to do with the matter of infallibility as the foolish Rafidis claim.

The Prophet (peace and blessings of Allah be upon him) said in other ahadith sahihah:

“Whoever obeys me had obeyed Allah and whoever disobeys me has disobeyed Allah. Whoever obeys my commander has obeyed me and whoever disobeys my commander has disobeyed me.” [Bukhari, no. 6718; Muslim, no. 1835]

According to consensus – even among the Rafidis – this does not mean that the commander is infallible; rather some of the commanders sent by the Prophet (peace and blessings of Allah be upon him) made mistakes in some matters which it is known went against the Book of Allah and the Sunnah of His Prophet (peace and blessings of Allah be upon him). An example of that is the proven report in which ‘Ali (may Allah be pleased with him) said:

“The Prophet (peace and blessings of Allah be upon him) sent out an expedition and he appointed in charge of them a man from among the Ansar and told them to obey him. He got angry and said: Didn’t the Prophet (peace and blessings of Allah be upon him) tell you to obey me? They said: Yes. He said: Then gather firewood for me. So they gathered it for him and he said: Light a fire. So they lit a fire. He said: Enter it. So they thought of entering it, but some of them held others back and said: We have fled to the Prophet (peace and blessings of Allah be upon him) to escape the fire. They stayed until the fire burned out and his anger died down. News of that reached the Prophet (peace and blessings of Allah be upon him) and he said: “If you had entered it you would never have come out of it until the Day of Resurrection. Obedience is only with regard to that which is right and proper.” [Bukhari, no. 4085; Muslim, no. 1840]

Hence the Prophet (peace and blessings of Allah be upon him) restricted this obedience only to that which is right and proper. So if the anger of Fatimah was part of the anger of the Messenger (peace and blessings of Allah be upon him), then according to consensus it is limited only to that which is right and proper, and if the anger of Fatimah was for reasons that were not in accordance with the law of Allah, then what is right and proper is to implement the law of Allah, even if it angered Fatimah. Hence the Prophet (peace and blessings of Allah be upon him) said:

“If Fatimah the daughter of Muhammad were to steal, I would cut off her hand.”

And he (peace and blessings of Allah be upon him) said:

“O Fatimah bint Muhammad, save yourselves, for I cannot avail you anything before Allah, ask for whatever you want of my wealth.”

See Manhaj al-Sunnah al-Nabawiyyah, 4/250.