What sort of object represents skyscaper sheaves on the etale site of $\mathbb{Z}_p$? By SGA 4 IX Proposition 2.7, any constructible sheaf $\mathcal{F}$ on a qcqs scheme $X$ can be represented as an equalizer of two etale maps between representable (by schemes) sheaves. This would imply, in particular, that the original sheaf is representable by an algebraic space that is etale over $X$.
What does the representing algebraic space look like if $X=\operatorname{Spec}{\mathbb{Z}_p}$, and $\mathcal{F}$ is the sheaf represented by $\operatorname{Spec}{\mathbb{F}_q}$? More generally, let's take a non-constant skyscraper sheaf represented by $\operatorname{Spec}{\mathbb{F}_q}$ for $q$ a power of $p$.
 A: Let $F$ be some constant étale sheaf on $Z=\operatorname{Spec} \mathbb{F}_p$. This is the constant sheaf $\Gamma_Z^\ast(S)$ for some set $S$. Let $X=\operatorname{Spec}(\mathbb{Z}_p)$, and let $E=\Gamma^\ast_X(S)$ be the corresponding constant sheaf on $X$.  This sheaf is representable by the scheme $X\times S=\coprod_{s\in S} X$ étale over $X$.
Then let $U=X\setminus Z$.  We can present $i_\ast(F)=\operatorname{colim}(U\times_X X\times S^2\rightrightarrows X\times S)$, which is an algebraic space because $U\times_X X\times S^2=U\times S^2\to U \to X$ is an étale scheme over $X$ as well, so the two arrows go between étale schemes over $X$ and therefore must be étale (a priori it is a DM stack, but the colimit is a sheaf rather than a stack).
(Jon Pridham just showed me this example earlier today!).
More generally for a local system, you can descend it to a local system on $\mathbb{Z}_p$ because it is Henselian with closed point given by $\mathbb{F}_p$, so their étale fundamental groups agree.  Then use the exact same construction as above.
Anyway, this is a completely general fact that every sheaf on the small étale site of a scheme $X$ is representable by some (potentially horrible) algebraic space that is étale over $X$ (note: I do not require any (quasi)separation or quasicompactness conditions on an algebraic space).
