The Picard iteration is widely used to find fixed points of locally contractive (LC) maps. This paper extends the Picard iteration to distributed settings; specifically, we assume the map of which the fixed point is sought to be the average of individual (not necessarily LC) maps held by a set of agents linked by a sparse communication network. An additional difficulty is that the LC map is not assumed to come from an underlying optimization problem, which prevents exploiting strong global properties such as convexity or Lipschitzianity. Yet, we propose a distributed algorithm and prove its convergence, in fact showing that it maintains the linear rate of the standard Picard iteration for the average LC map. As another contribution, our proof imports tools from perturbation theory of linear operators, which, to the best of our knowledge, had not been used before in the theory of distributed computation.