  In characteristic $p>0$ and for $q$ a power of $p$, we compute the number of
nonplanar rational curves of arbitrary degrees on a smooth Hermitian surface of
degree $q+1$ under the assumption that the curves have a parametrization given
by polynomials with at most $4$ terms. It is shown that a smooth Hermitian
cubic surface contains infinitely many rational curves of degree $3$ and $6$.
On the other hand, for all other cases the numbers of curves are finite and
they are exactly determined. Further such rational curves are given explicitly
up to projective isomorphism and their smoothness are checked.
