As far as a person’s sleeping after praying Fajr is concerned, no text (of Qur’an or hadeeth) has been reported to indicate that this is prohibited, so the general principle applies (i.e., everything is permitted except that which has been expressly forbidden).
But the practice of the Prophet (peace and blessings of Allah be upon him) and his Companions after praying Fajr was to remain sitting in the place where they had prayed until the sun rose, as is reported in.
It was narrated that the Prophet (peace and blessings of Allah be upon him) said: “Whoever sleeps after ‘Asr and loses his mind, let him blame no one but himself.” This hadeeth was classed as da’eef (weak) by Shaykh al-Albaani in al-Silsilah al-Da’eefah, 39.
Shaykh al-Albaani said:
Marwaan ibn Muhammad al-Asadi said: “I said to Layth ibn Sa’d – when I saw him sleep after ‘Asr during the month of Ramadaan – ‘O Abu’l-Haarith! Why are you sleeping after ‘Asr when Ibn Lahee’ah has told us … (and he mentioned it)?’ Al-Layth said: ‘I do not give up something that is good for me because of the hadeeth of Ibn Lahee’ah from ‘Aqeel!’”
I say, I am impressed by this answer of al-Layth, because it is indicative of understanding and knowledge, and no wonder, for he was one of the imams of the Muslims and well known fuqahaa’. I know that many of the shaykhs today do not let themselves sleep after ‘Asr, even if they need to do that. If it is said to them that the hadeeth concerning that is da’eef, they will immediately respond by saying, “Weak ahaadeeth should be followed with regard to virtuous actions.”
Think about the difference between the understanding of the salaf (earlier generation) and the knowledge of the later generations.
Al-Silsilah al-Da’eefah (hadeeth no. 39).