The earliest recorded source for this practice appears to be Tosefta: Yadayim 1:2:
הנוטל את ידיו צריך לשפשף את ידיו
[After] washing one's hands, one must rub one's hands.1
It's then cited, moree or less verbatim, by the Rambam in Hilchot Mikvaot 11:2 and by the Rema in Orach Chayim 162:2.2
Yodeyans msh210 and ezra reported in comments on this question that they personally implement this practice, msh210 reported having seen many others doing it, and ezra said that it's the typical practice in Lubavitcher communities. I think that's sufficient evidence to establish that at least some people practice it today.
(Regarding possible reasons for this practice, see Note #36 in Chabad.org's online presentation of Shulchan Aruch Harav 1623. Reasons cited include "additional purity," removing any remaining dirt, and ensuring that the washing-water spreads to all parts of the hands.)
1. My translation.
2. Thanks to DoubleAA for mentioning this source in a comment on the question.
3. Thanks to Oliver for mentioning this source and the Tosefta in comments on the question.