An element $x$ of a Hopf algebra $H$, is called a primitive element if $\Delta(x)=1\otimes x+x\otimes 1$. The set of primitive elements of $H$ is denoted $P(H)$. It can be shown that:

"If $H$ is a $\mathbb{k}$-Hopf algebra of finite $\mathbb{k}$-dimension and $\mathbb{k}$ is a field of characteristic zero, then $P(H)=\{0\}$"

The proof of the above fact, amounts to showing -inductively- that, if there is $0\neq x \in P(H)$ then $x^{n}$ (for all positive integers $n$) are linearly independent, which is a contradiction to the finite dimensionality of $H$.

Now the question: If we drop the requirement of characteristic zero (for the field $\mathbb{k}$ of coefficients) what are some examples violating the above proposition? In other words, considering finite dimensional hopf algebras over some finite field, let's say the group Hopf algebra of the cyclic group of order $2$ over a field of characteristic $2$, are there -explicit- examples of primitive elements?