Assalaamu alaykum, dear Scholar! I want to know whether the following hadith is authentic or not:
Prophet Muhammad, sallallaahu 'alayhi wa sallam, performed his first obligatory prayer in the cave of Hira in 610 in the night when the first revelation was revealed. Ibn Is-haak, may Allaah be pleased with him narrated:
“Prophet Muhammad, sallallaahu 'alayhi wa sallam, was in the upper part of Makkah. Gabriel, peace be upon him, appeared in front of him with his most beautiful face and his best smell and said the following, ‘O Muhammad! Allaah, The Exalted, has sent His salaams to you and has ordered me to convey that He has appointed you as His Messenger for the jinn and the people, so you will invite them to the word ‘Laa ilaaha illAllaah’. Then he hit his foot on the ground, and water started to gush out. Gabriel performed ablution with that water so that Prophet Muhammad would learn from him how to perform ablution in order to perform the prayer. Then he told Prophet Muhammad to perform ablution as he did. Thereafter, Gabriel performed a two-unit prayer that had four prostrations. Afterwards, the Messenger of Allaah took Khadijah, may Allaah be pleased with her, there, taught her to perform ablution, and performed two units of prayer with her.” (Ibn Hishaam, Ibn Jarir, Ibn Kathir)

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.

This story was mentioned by Ibn Hishaam in As-Seerah, quoting Ibn Is-haaq. Ibn Is-haaq said:

“Some scholars related to me that when the prayer was enjoined on the Messenger of Allah Jibreel (Gabriel) came to him in the upper part of Makkah. Jibreel hit the ground in the valley with his heel, so a spring of water gushed out, and Jibreel performed ablution while the Messenger of Allah was looking at him to learn the manner of ablution for the prayer. Then the Messenger of Allah performed ablution in the same manner as he saw Jibreel perform it. Then Jibreel stood and led him in prayer and then left. Afterwards, the Messenger of Allah came to Khadeejah, and he performed ablution in order to show her how to perform ablution for the prayer as Jibreel had shown him. So she performed ablution in the like manner as he had done, and then he led her in the prayer in the same manner as Jibreel led him in it.”

This story by Ibn Hishaam was mentioned in full in several sources quoted from Ibn Is-haaq on the authority of the scholars who narrated it to him. However, the story with this chain of narrators is interrupted as the person who narrated it to Ibn Is-haaq is unknown, let alone that he did not live at the time of the Prophet . Nevertheless, the story is strengthened by other narrations, including the hadeeth of Zayd ibn Haarithah, who related from the Prophet that Jibreel, may Allah exalt his mention, came to him at the beginning of the revelation and taught him ablution and the prayer. This hadeeth was reported by Ahmad, Ibn Maajah and others, and Al-Albaani classified it as hasan (good).

Al-Bayhaqi said in Dalaa’il An-Nubuwwah (Signs of Prophecy):

“Ibn Lahee’ah narrated from Abul Aswad, from ’Urwah ibn Az-Zubayr, this story in a similar wording, and added, ‘Jibreel, may Allah exalt his mention, opened a spring of water and performed ablution while Muhammad was looking at him; so he washed his face and hands to the elbows and wiped his head and washed his feet to the ankles, then he splashed water on his private parts and then prostrated two prostrations facing the House (the Ka'bah), so Muhammad did the same thing as he saw Jibreel doing.’”

However, we have not come across any narration where it says that this prayer took place in the cave of Hiraa' on the date that you mentioned in the question.

Allah knows best.