First some definitions, since the notation is somewhat non-standard: let $\mathcal P(m,n)$ denote the vector space of polynomials with real coefficients of degree at most $m$ in $n$ variables. Let $E \subset \mathbb R^n$ be closed, and let $P_\bullet: E \to \mathcal P(m,n)$. When we write $P_x(y)$, we mean $x \in E$ and $y \in \mathbb R^n$. Let $\||P_\bullet\||_{K,\alpha} = \sup\limits_{x \in K} |(\partial^\alpha |_{y=x} P_x(y))|$, where $\alpha$ is a multi-index and $K \subset E$ is compact. Let $$R_\alpha(x,y) = {\partial^\alpha P_x(x) - \partial^\alpha P_y(x) \over |x-y|^{m-|\alpha|}}$$ when $x \neq y$ and $R(x,y)=0$ when $x=y$, and let $\||P_\bullet\||_{K, \dot \alpha} = \sup\limits_{x,y \in K} |R_\alpha(x,y)|$. Let $C^m(E) = \{P_\bullet: E \to \mathcal P(m,n) :$ $ \||P_\bullet\||_{K, \alpha}< \infty, \||P_\bullet\||_{K, \dot \alpha}<\infty, R_\alpha \in C(E), \forall |\alpha|$ $ \leq m, \forall K \subset \subset E\}$.
The space $C^m(E)$ admits a ring structure: let $$(P \star Q)_x(y) = P_x(y) Q_x(y) / ((y-x)^\alpha: |\alpha|=m+1).$$ There is a map $J^m_E : C^m(\mathbb R^n) \to C^m(E)$ (where $C^m(\mathbb R^n)$ is the usual $C^m$) given by letting $J^mF_x$ be the $m$th order Taylor polynomial of $F$ at $x$. This map is a ring homomorphism, by the product rule. Whitney's extension theorem states that the map $J^m$ admits a continuous, linear right inverse. This is why I am calling this space $C^m(E)$.
I would like to know when $C^m(E)$ admits nilpotents, i.e. when does there exist non-zero $P_\bullet \in C^m(E)$ such that the constant terms of $P_x$, factored at $x$, are zero for all $x \in E$. Here are two examples. Let $n = m = 1$, and let $E = \{0\}$. Let $P_0(x) = x$. It is trivial to see that $P_\bullet \in C^1(E)$ (there is basically nothing to check), and $P_\bullet$ is nilpotent: $P_\bullet^2=0$. On the other hand, let $E \subset \mathbb R^n$ have dense interior. If $g(x) = P_x(x)$, and $P_\bullet \in C^m(E)$, then $g$ is $C^m$ (in the usual sense) on the interior of $E$. Therefore, if $P_\bullet$ is nilpotent, $g$ vanishes. But since the interior of $E$ is dense, this implies that $P_\bullet$ vanishes.
(Some of these tags are kind of guesses, I hope they are not inappropriate.)