Do Kippot and Keffiyehs Share an Etymology?

The novelist Dara Horn participates in daf yomi, the worldwide daily-page-of-Talmud-study program. This prompts her to ask:

In the tractate of Sanhedrin recently, we came to a discussion of what the Mishnah calls a kippah, a domed or vaulted chamber in which convicted criminals were kept. In considering the opinions of various commentators about this now-obsolete use of the Hebrew word, I found myself thinking of its better-known meaning of “skullcap” and sometimes “hood” (as in kippah adumah, Red Riding Hood), as well as that of “dome” (as in kippat barzel, the Iron Dome anti-missile system). From there my mind, no doubt influenced by ongoing pro-Palestinian and anti-Israel activism in the United States, wandered to the Arabic word keffiyeh and to my question for you. Might keffiyeh be a cognate of kippah?

As sister Semitic languages, Hebrew and Arabic do share a large number of cognate words, but kippah-keffiyeh would not seem to be among them. Kippah first occurs in the Mishnah, where it signifies both a dome and a skullcap (although it’s obvious why one should suggest the other, it’s unclear which suggested which); it derives from the Hebrew verb kafaf, to bend, its root meaning being of something bent or rounded over something else. Arabic has no corresponding verb or noun. It does have two nouns with Hebrew cognates that are related to kafaf and kippah: kaf or kaffa, “palm of the hand,” the same as the Hebrew word for palm, kaf or kaf-yad (palms form domes when hands are bent), and akaf, “saddle blanket,” akin to Hebrew ukaf, “saddle.” Neither of these words, however, can have anything to do with keffiyeh, nor does Arabic have other words that might.