Not by the name of Mamluk but by the name of Ghulam many Islamic states used them. Mamluk name was later emerged and took place of ghulam in Mamluk Empire.

Ghulam system was invented by the Abbasid Caliph Al-Mutasim to make use of the horsemanship of Turkic nomads. Though before Abbasids in Umayyads there were such elements too. In the year 674 for example Basra governor/commander of Muaviye bin Ebu Sufyan, Ubeydullah bin Ziyad brought 2000 Turkic archers and put them in Basra.

In fact many islamic states were founded by ghulams. Delhi Sultanate in India, Mamluk Sultanate, Ghaznavids, Khwarezmians, Tolounids of Egypt, Ikhshidites of Egypt. These states were founded by ghulams/mamluks and continued the practice.

Apart from them Seljuks also used ghulams.

Great Seljuk Empire would take the pagan Turkic people, convert them to islam and raise them as elite soldiers. But its offshot in Anatolia: Anatolian Seljuk Sultanates ghulams were mostly hellenized Anatolians or Armenians because there werent many pagan Turks so their pool was limited to Christians of Anatolia.

After a certain point all Turkic people basically islamified and since it was a sin to enslave muslims in shariah law now Christians of Caucasus and Balkans were preferred. Thats how janissaries were born in Ottoman Empire. In Mamluk Empire in the first half Kipchak Turks were recruited but in the reign of Sultan Qalawun who himself was also a Kipchak, started recruiting Circassians to counterbalance Turkic Mamluks political power.

North African kingdoms from what I know didnt have ghulams. Mongols or Mongol states in middle east didnt inherit ghulam system but Aq Koyunlus, a Turkmen state did have ghulams. Their successor Safavids also did. I didnt hear it for the Afsharids but Qajars also had ghulams. Safavids disnt have ghulams until Shah Abbas period but Shah Abbas was so sick of inner Turkmen clans political fights that he decided to form ghulams among Caucasian Christians again. Mainly Georgians.