I have traced back this claim to its original source. Come along with me on this journey! Let's start with a source which used to be linked on Wikipedia and is commonly cited online:
For ten miles in a direct line on the darkest night the pedestrian could walk securely through the city and its environs by the light of innumerable lamps.
Samuel Parsons Scott, History of the Moorish Empire in Europe (1904), vol. 1, p. 619.
This book has a massive bibliography but no footnotes, which makes it useless. However, it did make me google harder, and find an 1833 reference:
For ten miles the citizens could travel by the light of lamps along an uninterrupted extent of buildings.
Andrew Crichton, The History of Arabia (New York: Harper, 1845 ), vol. 2, 52
The only source cited anywhere near this sentence is Washington Irving (!), who wrote a book called Chronicle of the Conquest of Granada in 1829. I found that this section of Irving's book was derived from an even earlier book:
Ashshakandy† relates, in one of his works, that through Cordova, with the continuations of Azzahra and Azzahira, he had travelled ten miles by the light of lamps along an uninterrupted extent of buildings.
† Abū'l Walid Ismāil, an eminent man from Shakanda, which is an ancient town on the south side of the river over against Cordova. He wrote a treatise on the excellence of Andalusia, in opposition to Abū Yahya, who vindicated that of Africa; and died at Seville in the year 629.
John Shakespear and Thomas Hartwell Horne, The History of the Mahometan Empire in Spain (London: William Bulmer and Co., 1816), 162
This incredibly helpful footnote points us to the original source, a 13th century CE Andalusian named aš-Šaqundī!
Se dice que estaban tan pobladas las construcciones de Córdoba, al-Zahrā’ y al-Zāhira, que se podía caminar por ellas a la luz de las lámparas por espacio de diez millas, sin interrupción alguna. (Google Translate: "It is said that the buildings of Córdoba, al-Zahrā’ and al-Zāhira, were so populated that one could walk through them by the light of the lamps for ten miles, without any interruption.")
Abu-l-Walid Ismāʿīl Ibn-Muḥammad aš-Šaqundī, Elogio del islam español (Risāla fi fadl al-Andalus), trans. Emilio García Gómez (Madrid: Impr. de E. Maestre, 1934), p. 105
Finding this original source corrects a major error introduced through the chain of transmission and helps us understand the type of lighting being discussed here. These were not street lamps installed by the caliph. This would not have been a system of lamps posted on the road, but lamps installed outside doors by homeowners -- or, as commenters suggest, lamps from the inside of homes.
We actually have examples of this exact model of Andalusian outdoor lamp in museums. You can see the spike inside on which a tallow candle would have been placed. The shape of the lamp would allow users to post it directly over a door.
There is also a good article about Andalusian indoor lighting online which shows that indoor lamps of that time used oil and a fuel source resting on top of water in a glass fixture. Some of the Islamicate glass fixtures of this time are noted for their beauty.