  We prove that the weighted composition operator $W_{\phi,\varphi}$ fixes an
isomorphic copy of $\ell^p$ if the operator $W_{\phi,\varphi}$ is not compact
on the derivative Hardy space $S^p$. In particular, this implies that the
strict singularity of the operator $W_{\phi,\varphi}$ coincides with the
compactness of it on $S^p$. Moreover, when $p\neq2$, we characterize the
conditions for those weighted composition operators $W_{\phi,\varphi}$ on $S^p$
which fix an isomorphic copy of $\ell^2$ .
