We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf E\mathcal EE of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an E\mathcal EE appears as an extension of two Lehn–Lehn–Sorger–van Straten sheaves. Then we prove that a general deformation of E(1)\mathcal E(1)E(1) becomes Ulrich. In particular, this says that general cubic fourfolds have Ulrich complexity 6.