In this work, we investigate the positive solutions of a semilinear nonlocal diffusion equation with power nonlinearity. We have discovered a new phenomenon that the equation at the critical exponent, under a certain condition on the kernel function, admits a global in time solution for a small initial datum. This is in sharp contrast with the corresponding heat equation and the nonlocal diffusion equation with a regular or fractional Laplacian kernel, where every (nontrivial) positive solution always blows up in finite time at the critical exponent.