Is the following statement true?......
If $C$ is a non-degenerate closed and connected subset of the Euclidean plane $\mathbb R ^2$ and $p$ is any point of $C$, then there exists a connected subset of $C$ which contains p and has an arbitrarily small positive diameter.
If the answer is "yes", is this statement still true for any finite dimensional Euclidean space?
Yes, and yes it also holds in any finite dimensional Euclidean space.
Let $C$ be a closed connected subset of $\mathbb R ^n$.
Then $C$ has a compactification $\gamma C$ (such as $\text{cl}_{[-\infty,\infty]^n} C$), and will be open in $\gamma C$ because it is locally compact.
Let $c\in C$ and let $U=B(c,\epsilon)$ be an $\epsilon$-neighborhood of $c$. Note that $U$ is also open in $\gamma C$. By the boundary bumping theorem for connected compact Hausdorff spaces, the component $K$ of $c$ in $U$ meets the boundary (in $\gamma X$) of $U$. Then $K$ is nontrivial, $c\in K\subseteq C$ and $K$ diameter less than $2\epsilon$.
Note: You do not really need $C$ to be closed; it could be open also. The important thing is that it is locally compact. You could have a big problem if $C$ is not locally compact. Consider the Knaster-Kuratowski fan in $\mathbb R ^2$. No point other than the dispersion point has connected sets around it of small diameter.