In the present paper, we would like to draw attention to a possible generalized Fisher information that fits well in the formalism of nonextensive thermostatistics. This generalized Fisher information is defined for densities on $\mathbb{R}^{n}.$ Just as the maximum Rényi or Tsallis entropy subject to an elliptic moment constraint is a generalized q-Gaussian, we show that the minimization of the generalized Fisher information also leads a generalized q-Gaussian. This yields a generalized Cramér-Rao inequality. In addition, we show that the generalized Fisher information naturally pops up in a simple inequality that links the generalized entropies, the generalized Fisher information and an elliptic moment. Finally, we give an extended Stam inequality. In this series of results, the extremal functions are the generalized q-Gaussians. Thus, these results complement the classical characterization of the generalized q-Gaussian and introduce a generalized Fisher information as a new information measure in nonextensive thermostatistics.