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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 $p$.

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  • $\begingroup$ This is tautological. We define Spaces by declaring all sheaves to be Spaces. Then a sheaf is its own espace etale (though there is a finiteness condition to check in the claim that it is an Algebraic Space). . . . Look at more examples of algebraic spaces. Seek more examples; maybe try Knutson. Anyhow, example 2 is the answer to your question (mutatis mutandis). $\endgroup$ – Ben Wieland Jul 14 '20 at 0:15
  • $\begingroup$ @Ben: Yes, I know, but the question is, what kind of space? What properties does it have? What's an explicit description? $\endgroup$ – David Corwin Jul 14 '20 at 22:21
  • $\begingroup$ Did you follow my link? $\endgroup$ – Ben Wieland Jul 15 '20 at 0:24
  • $\begingroup$ Oops sorry - now I see. Thanks! $\endgroup$ – David Corwin Jul 16 '20 at 2:24
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    $\begingroup$ $F_p$ reps terminal sheaf. $i_*$ rep by $Z_p$, also terminal. First study skyscraper sheaves in the usual topology or in the Zariski topology. Push forward the coproduct of two copies of the terminal sheaf, represented by the disjoint union of two copies of $\mathrm{Spec}F_p$ produces something weirder. It is represented by "the line with the doubled origin," a scheme where 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. $\endgroup$ – Ben Wieland Jul 16 '20 at 12:58
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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).

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