Not from chalk in the ground: most likely from Chalcot, 'cold cottage'.
The area stands on London Clay. Chaldecote, a common English place name meaning cold cottage, is recorded here in 1256; it had become Chalcotts by the 16th century and Chalk by 1746, and in 1786 the estate was commonly known as Chalk Farm. Others argue that the farm was never called Chalcot and was probably named after its whitewashed buildings, as chalk was used in whitewash.