Is the hadith narrated on the authority of Aisha – may Allah be pleased with her – in which she describes Sawdah and praises her by saying that she (Sawdah) granted her (Aisha) her day, then Aisha said: "However, she has some envy," authentic, and is this statement not considered backbiting?

The aforementioned Hadith (definition) is authentic, without the phrase "except that there is envy in it." Muslim narrated it with the wording: "Aisha (may Allah be pleased with her) said: 'I have never seen a woman more beloved to me to be in her skin than Sawdah bint Zam'ah, a woman with keenness.' She said: 'When she grew old, she gave her day from the Messenger of Allah (peace and blessings be upon him) to Aisha.' She said: 'O Messenger of Allah, I have given my day from you to Aisha.' So, the Messenger of Allah (peace and blessings be upon him) used to allot two days for Aisha: her day and Sawdah's day."

There is nothing in the Hadith that constitutes backbiting. Aisha is praising Sawdah. Her saying "in her skin" means in her very being, and "a woman with keenness" is for clarification. Aisha did not intend to criticize Sawdah by this, but rather described her with strength of character and excellent intellect.