Choose d+1 distinct residue-field elements and lift them to binary coordinate vectors a_0,...,a_d in O. Their pairwise differences are units. Define L_i(T)=product_{j!=i}((T-a_j)/(a_i-a_j)). This polynomial is one at a_i and zero at every other a_j. For a polynomial P of degree at most d, P(T)=sum_i P(a_i)L_i(T). For uniqueness, a difference vanishing at all points is successively divisible by each monic T-a_i: unit point differences ensure each remaining factor still vanishes at the other points. Its degree is too small unless it is zero. This works over A_p despite zero divisors. Without unit differences it can fail: modulo 4, the nonzero linear polynomial 2T vanishes at both 0 and 2.