1520s, "pointed top, projecting summit," a variant of pike (n.4) "sharp point." Meaning "top of a mountain, a precipitous mountain with a more or less conical summit" is recorded by 1630s, though pike was used in this sense c. 1400. Figurative sense is 1784. Of beards, 1590s; of hats, 1650s. Meaning "point formed by hair on the forehead" is from 1833. As "the highest point" in any varying quantity, or the time when this occurs, by 1902.
The Peak, the prominent hill in Derbyshire, England, is older than the word for "mountaintop;" compare Old English Peaclond, for the district, Pecsaetan, for the people who settled there, Peaces ærs for Peak Cavern. In this case it is sometimes said to be a reference to an elf-denizen Peac "Puck."
1570s, "to rise in a peak," from peak (n.). Figurative meaning "reach the highest point" is recorded by 1958. Related: peaked; peaking.