Assalaamu alaikum. Are we allowed to keep hamster as pets?

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.

The animal that you mentioned in the question –hamster –is mentioned in the Arabic Encyclopedia and the Free Encyclopedia, and it is known in the Arabic language by Qidaad.
In this Encyclopedia, they put a picture for it, which shows that it is very similar to a mouse. This strengthens the belief that it is one of the genres of mice, especially that we know that some writers mentioned that there are studies which showed that there are 86 genres of mice, that includes 720 species [kinds].

Since the hamster is from the genre of mice, then it is not permissible to buy it or keep it as a pet, because the Sunnah stipulated that the mouse be killed whether inside or outside the boundaries of al-Haram (Makkah), as in the Hadeeth narrated by 'Aa'ishah that the Prophet said: “There are five harmful animals which should be killed outside or inside the sacred areas (the Haram): the scorpion, the kite, the mouse (rat) , the crow, and a biting dog.” [Al-Bukhari]

Ibn Hajar said in Fath al-Baari when explaining this Hadeeth: “There are many kinds of mice, including the rat, the mole,…….[he mentioned three other kinds], the ruling regarding the prohibition of eating them and the permissibility of killing them is the same.” [End of quote]

Allah knows best.