The Bass-Quillen conjecture is known to be true for principal ideal domain (that is, if $A$ is a PID, all finitely generated projective modules over $A[T_1,\dots,T_n]$ are free). This was proven in theorem 4 of the paper

<cite authors="Quillen, Daniel">_Quillen, Daniel_, [**Projective modules over polynomial rings**](http://dx.doi.org/10.1007/BF01390008), Invent. Math. 36, 167-171 (1976). [ZBL0337.13011](https://zbmath.org/?q=an:0337.13011).</cite>