The authors consider a market with an asset price described by fractional Brownian motion, which can be traded with temporary nonlinear price impact. The asymptotically optimal strategies for the maximization of expected terminal wealth are obtained. These strategies generate an average terminal wealth that grows with a power of the horizon, the exponent depending on both the Hurst and the price-impact parameters. Obtained results are extended to long memory Gaussian processes and to a class of H-self-similar processes.