In this article, we consider fractional derivatives of local time for $d-$dimensional centered Gaussian processes satisfying certain strong local nondeterminism property. We first give a condition for existence of fractional derivatives of the local time defined by Marchaud derivatives in $L^p(p\ge1)$ and show that these derivatives are Hölder continuous with respect to both time and space variables and are also continuous with respect to the order of derivatives. Moreover, under some additional assumptions, we show that this condition is also necessary for existence of derivatives of the local time with the help of contour integration.