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
Quillen, Daniel, Projective modules over polynomial rings, Invent. Math. 36, 167-171 (1976). ZBL0337.13011.