  We show that every non-compact weighted composition operator $f \mapsto
u\cdot (f\circ\phi)$ acting on a Hardy space $H^p$ for $1 \leq p < \infty$
fixes an isomorphic copy of the sequence space $\ell^p$ and therefore fails to
be strictly singular. We also characterize those weighted composition operators
on $H^p$ which fix a copy of the Hilbert space $\ell^2$. These results extend
earlier ones obtained for unweighted composition operators.
