Is the embedding of $W^{2,p}$ onto $C^1(\overline{I})$ compact?

We know that when $$I$$ is a bounded interval and $$1 that the injection $$W^{1,p}\subset C(\overline{I})$$ is compact.

The proof of this fact uses the Arzela-Ascoli theorem on the unit ball $$\mathcal{H}$$ of $$W^{1,p}(I)$$. Is it true that the embedding of $$W^{2,p}$$ onto $$C^1(\overline{I})$$ is also compact?

Suppose that $$u\in W^{2,p}$$ then $$u'\in W^{1,p}$$ and thus $$u'\in C(\overline{I})$$ and therefore $$u\in C^1(\overline{I})$$.
Now let $$\mathcal{H}$$ be the unit ball in $$W^{2,p}$$. Then $$\mathcal{H}$$ is equicontinuous since for all $$u\in \mathcal{H}$$, $$|u(x)-u(y)|=\left|\int_{y}^xu'(t)dt\right|\leq \|u'\|_p|x-y|^{1/p'}\leq |x-y|^{1/p'}.$$ Hence by Ascoli-Arzela, $$\mathcal{H}$$ has compact closure in $$C(\overline{I})$$. Since $$\mathcal{H}\subset C^1(\overline{I})\subset C(\overline{I})$$ then $$\mathcal{H}$$ has compact closure in $$C^1(I).$$ Is there anything wrong with the argument?