It is essential to verify stories before narrating them and spreading them
Question

In high school, some of the teachers tell us stories about the angels and about Sayyiduna Jibreel (peace be upon him), and they said that Allah, may He be glorified and exalted, commanded Jibreel to take the soul of a woman (in death), so he went to her and found her breastfeeding her infant, so he could not take her soul. He came back three times, and on the last occasion he took her soul. And there are other stories. Are these stories based on reality, or is there no sound basis for them?
Praise be to Allah.Firstly:
The Muslim should verify what is told to him, and what he himself wants to narrate. So he should not accept anything but what is proven, and he should not narrate anything except what he knows to be sound and true.
People find tales and stories exciting, but most of these stories are lies.
Muslim narrated in the Introduction to his Saheeh (1/10) that Abu Hurayrah said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “It is sufficient lying for a man to speak of everything that he hears.”
An- Nawawi (may Allah have mercy on him) said concerning this and similar hadiths:
This is a warning not to speak of everything that one hears, for usually a person hears things that are true and things that are false, so if he were to speak of everything that he hears, he would be telling lies, because he would be telling of something that did not happen.
End quote from Sharh Saheeh Muslim (1/75)
See also question no. 14212.
Secondly:
The angels are part of the world of the unseen, and we should not speak of the unseen except in terms of the sound reports that are proven to be true. As for reports that are not sound (saheeh), it is not permissible to speak of them with regard to matters of the unseen. So what is required of us is to believe in the angels, and to believe in the unseen, and to believe that the unseen belongs to Allah, and we have no right to speak of the unseen except in terms of what is proven in the religious texts.
See question no. 843.
Thirdly:
The one who is charged with taking people’s souls in death is the Angel of Death (peace be upon him), not Jibreel (peace be upon him). Rather Jibreel is charged with bringing the revelation.
See: Fath al-Baari (6/307).
See also question no. 40671 .
Fourthly:
As for the story referred to, we know of no basis for it in the books of Ahl as-Sunnah. The story sounds odd, for when Allah commands the Angel of Death to take a person’s soul, he takes it without any hesitation, delay or prevarication; he is not more merciful towards people than their Creator, and he is not kinder to them than the Lord of the Worlds, the Most Gracious, Most Merciful. He is far above the lies of the fabricators and the false words of those who make things up.
The idea that he went to that woman to take her soul, and found her breastfeeding her infant, so he could not take her soul, is false on two counts:

Allah, may He be exalted, is more merciful to her and her child than the Angel of Death, so how can it be said that the Angel of Death did not take her soul out of mercy towards her or her infant?

It was narrated from ‘Umar ibn al-Khattaab that he said: Some prisoners were brought to the Messenger of Allah (blessings and peace of Allah be upon him), and there was a woman among the prisoners who was searching for someone. When she found a small boy among the prisoners, she clasped him to her and started to breastfeed him. The Messenger of Allah (blessings and peace of Allah be upon him) said to us: “Do you think that this woman would throw her child into the fire?” We said: No, by Allah, she would never do that if she is able not to. The Messenger of Allah (blessings and peace of Allah be upon him) said: “Allah is more merciful towards His slaves than this woman is towards her child.” Narrated by al-Bukhaari (2754) and Muslim (2754).

The Angel of Death, and all the other angels of Allah, may He be exalted, do what they are commanded, without any hesitation or delay, and they do not disobey Allah for even the blink of an eye, as He, may He be exalted, says (interpretation of the meaning):

“They do not disobey Allah in what He commands them but do what they are commanded”
[at-Tahreem 66:6].
That is, no matter what He, may He be exalted, commands them to do, they hasten to do it and do not delay for even a blink of an eye.
End quote from Tafseer Ibn Katheer (8/168).
So beware of these stories, narrations and reports that have no basis, and no one knows where they came from, or who narrated them, or whether they are sound or not.
Rather the Muslim should focus on the reports and stories that are sound, not those that are not sound, let alone those that have no basis. In that which is sound there is sufficient blessing.
And Allah knows best.