For more information, I would recommend reading "Paul and Gamliel", in Jacob Neusner and Bruce Chilton (eds.), In Quest of the Historical Pharisees (Baylor University Press, 2007), 175-223 - but especially p208 onwards.
Short answer: nobody named Paul (nor, for that matter, Shaul) ever gets mentioned as a student of this or any other Gamliel, although some teachings associated with Gamliel might be viewed as an influence over some Pauline passages in the NT epistles.
A few caveats: note that growing up in the dust at someone's feet does not necessarily equal being a disciple, and people may have taken more disciples than those that get named.
Secondly, the rabbinic corpus has a distinct anti-sectarian tendenz, so even if Paul really were a disciple of Gamliel's it is unlikely that he would be so named.
And thirdly, there is also a well-attested phenomenon whereby a convert or a penitent might exaggerate the nature of their former religiosity (or lack thereof). Christian scholars generally assume Paul's claim in this regard to be one such example of an exaggeration. Note that Gamliel is nowhere else mentioned by Paul - not even in his epistle to the Philippians, where he claims to have once been a Pharisee.