Is $\mathbb{Z}_p [x]$ finite? Seems to me like it is.  There are only finitely many distinct powers of $x$ modulo $p$, by Fermat's Little Theorem (they are $\{1, x, x^2, ...,  x^{p-2}\}$), and the coefficient that I choose for each of these powers can only be taken from $\{0,1,2,..., p-1\}$.  So essentially I'm choosing amongst $p$ things $p-1$ many times, resulting in at most $p^{p-1}$ distinct polynomials. 
Yet an assignment claims that $\mathbb{Z}_p[x]$ is infinite. 

$\Bbb Z_p[x]$ is indeed infinite, since it contains polynomials, with coefficients in $\Bbb Z_p$, of arbitrary high degree, e.g. $x^n \in \Bbb Z_p[x]$, where $n \in \Bbb N$; also, $x^n \ne x^m$ if $m \ne n$.
Fermat's Little Theorem, that $a^p = a$ for $a \in \Bbb Z_p$, does not apply to the indeterminate $x \in \Bbb Z_p[x]$.  $x \in \Bbb Z_p[x]$ and $a \in \Bbb Z_p$ are definitely "birds of a different feather" in this regard.
So though $\Bbb Z_p$ is finite, $\Bbb Z_p[x]$ is not.
