  In this paper we prove that the $\mathcal{E}^\dagger_K$-valued cohomology,
introduced in [9] is finite dimensional for smooth curves over Laurent series
fields $k((t))$ in positive characteristic, and forms an
$\mathcal{E}^\dagger_K$-lattice inside `classical' $\mathcal{E}_K$-valued rigid
cohomology. We do so by proving a suitable version of the p-adic local
monodromy theory over $\mathcal{E}^\dagger_K$, and then using an \'{e}tale
pushforward for smooth curves to reduce to the case of $\mathbb{A}^1$. We then
introduce $\mathcal{E}^\dagger_K$-valued cohomology with compact supports, and
again prove that for smooth curves, this is finite dimensional and forms an
$\mathcal{E}^\dagger_K$-lattice in $\mathcal{E}_K$-valued cohomology with
compact supports. Finally, we prove Poincar\'{e} duality for smooth curves, but
with restrictions on the coefficients.
