While I don't know whether Rashi meant this exactly, there were definitely Arabs at the time, i.e., there were nomadic tribes settled in Arabia that may be considered some of the ancestors of modern Arabs. Rabbi Philip Biberfeld in Universal Jewish History, vol. 1, pg. 92 calls them “pure Arabs”. These ancient Arabs were descendants of Yoktan, one of the sons of Ever. Rabbi Ahron Marcus in Barzilai, pg. 148-149 identified them with the Arabian tribe Kakhtan, a group of Southern Semites who travelled further south from Mesopotamia to Arabia. Furthermore, in his book Keset Hasofer (the last surviving bit of his commentary on Tanach), pg. 303, he explains that the modern Arabs are descendants of two distinct ethnic Arabian groups: The sons of Yoktan (the Kakhtan) and the sons of Yishmael.
Early Islamic historians also identified Yoktan/Kakhtan as the ancestor of the Southern Arabians, as did Rasag in his tafsir on Beresheet 10:26 (while Yishmael is the ancestor of the Northern Arabians). The names of the sons of Yoktan refer to areas in which they settled in and/or names of sub-tribes. Rasag even translated the places they settled in as referring to Mecca, Madinah and other cities in Arabia (see here and here as well).
On pg. 285 of Keset Hasofer, Rabbi Marcus identifies Chavilah (one of the descendants of Kush) as an area within Arabia; Savtakh (Savta, another descendant) as a city in Arabia mentioned by Strabo (and see here); and Sheva (a third descendant) as a tribe that settled in Arabia, in an area bordering on the Red Sea.