The title pretty much summarizes the question: does every $p$-group have a *faithful* unipotent representation (with coefficients in $\mathbb{F}\_p$ or some finite extension thereof)? 



faithful unipotent representations of (finite) $p$-groups
I guess you mean finite $p$-groups. Then every finite-dimensional representation of a finite $p$-group $G$ is unipotent in characteristic $p$. (Indeed, all eigenvalues of every element of $G$ are $p$-power roots of unity and therefore must be equal to $1$. This means that every element of $G$ acts as a unipotent operator.) An example of a faithful unipotent representation of $G$ is provided by its regular representation over any field of characteristic $p$. In particular, the regular representation of $G$ over the prime field $F\_p$ is faithful and unipotent.


(The same construction works for infinite $p$-groups $G$ as well. Of course, in this case the ``regular" representation space of functions on $G$ with finite support will be infinite-dimensional.)


