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).

schemewhere two copies of $\mathrm{Spec}Z_p$ are glued along $\mathrm{Spec}Q_p$. The full equivalence relation is: $\mathrm{Spec}(Z_p\times Q_p)\Rightarrow\mathrm{Spec}(Z_p\times Z_p)$. $F_q$ yields etale twisted version. $\endgroup$2more comments