The earliest textual foundation appears in Mishnah Ketubot 7:6, which lists among the violations of "dat Yehudit" (Jewish law of modesty) a woman who goes out with her head uncovered and who spins in the marketplace — establishing that public exposure of the body is a recognized breach of Jewish modesty norms.
Building directly on that Mishnah, the Rambam (Mishneh Torah, Marriage 24:12) specifies that dat Yehudit — the customary modesty standard of Jewish women — is violated when a woman spins in the marketplace and "shows her arms to people" (מַרְאֵית זְרוֹעוֹתֶיהָ לִבְנֵי אָדָם), making the exposure of a woman's arms in public an explicit classical source.
The Shelah (Shenei Luchot HaBerit, Shaar HaOtiyot, Tzniut:1) extends this principle broadly, ruling that a woman is obligated to be maximally modest and that nothing from her body should be exposed even minimally, lest anyone stumble through what they see.
The broader framework grounding all these rules is articulated by the Arukh HaShulchan (Even HaEzer 21), who grounds the obligation of distance from sexual impropriety in the command of "kedoshim tihyu" (Vayikra 19:2), and notes that since a person's desires are drawn toward such matters, many fences and safeguards are required — including the prohibition against gazing at women whose limbs are uncovered.