Is a forty-day spiritual retreat (chilla) part of Islam?

Yes, it has a basis. The Prophet ﷺ said whoever worships Allah sincerely for forty days will find fountains of wisdom flowing from his heart, and he himself withdrew to the cave of Hira for worship.

In many Sufi paths, a seeker spends a period, often forty days, in focused seclusion and worship. This rests on the words of the Prophet ﷺ: “Whoever worships Allah for forty days with sincerity, fountains of wisdom shall gush forth from his heart and from his tongue.”

And there is the Prophet's ﷺ own example before revelation began: he became devoted to seclusion in the cave of Hira, worshipping for nights on end. Time set aside for concentrated worship is not foreign to Islam; it is drawn from its very origins.