In a market with an asset price described by fractional Brownian motion, which can be traded with temporary nonlinear price impact, we find asymptotically optimal strategies for the maximization of expected terminal wealth. Exploiting the autocorrelation in increments while limiting trading costs, 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. The resulting Sharpe ratios are bounded, insensitive to the horizon, and asymmetric with respect to the Hurst exponent. These results extend Gaussian processes with long memory and to a class of self-similar processes.