Another reasonable continuation for white after 3. ... b5 is:
1.d4 d5 2.c4 dxc4 3.Nf3 b5 4.a4 c6 5.e3
This transposes into a position more commonly reached via the Slav move order. 1.d4 d5 2.c4 c6 3.Nf3 dxc4 4.e3 b5 5.a4.
Here you can find a lot more example games to look through. But still Black does not hold onto the pawn and instead gives it back. The most standard continuation here is:
1.d4 d5 2.c4 c6 3.Nf3 dxc4 4.e3 b5 5.a4 e6 6.axb5 cxb5 7.b3 Bb4+ 8.Bd2 Bxd2 9.Nbxd2 a5 10.bxc4 b4
White has recovered the pawn. Stockfish rates this position as a clear advantage for white (+1.23), though Shredder's grandmaster database has it dead even with white only doing well after 11.Ne5 and with black doing better after Stockfish's suggested 11.Bd3. So it seems that white has the advantage here but that black has good practical chances.