At the prime 2, Behrens, Hill, Hopkins and Mahowald showed that M-2 (1, 4) admits a 32-periodic v(2)-self-map. More recently, in joint work with Mahowald, we showed that A(1) also admits a 32-periodic v(2)-self-map. This leads to the question of whether there exists a finite 2-local complex with periodicity less than 32. We answer this question in the affirmative by producing a class of finite 2-local spectra (Z) over tilde all of which admit a 1-periodic v(2)-self-map. (C) 2019 Elsevier Inc. All rights reserved.