The comments are correct that because $\mathbb{F}_p \in D^{\mathrm{perf}}(\mathbb{Z})$ the Barr-Beck-Lurie applies. But we can prove the analogous result for the residue field of any prime $\mathfrak{p}$ in any Noetherian ring $A$, without assuming $\mathfrak{p}$ is generated by a single regular element. That is, the derived extension of scalars functor from $\mathfrak{p}$-local, $\mathfrak{p}$-complete complexes to $D(\kappa(\mathfrak{p}))$ is comonadic.

We can assume wlog that $A$ is a local ring with maximal ideal $\mathfrak{p}$. Let $K$ be the koszul complex on a list generators for $\mathfrak{p}$. Then $K$ is dualizable and by Proposition 3.34 of "The Galois group of a stable homotopy theory" the map $K \to k$ admits descent (in the strong sense of Akhil Mathew); in particular $D(K) \to D(\kappa(\mathfrak{p}))$ is comonadic. Dualizability of $K$ means that $D^{\mathfrak{p}-\mathrm{comp}}(A) \to D(K)$ preserves all limits, hence is comonadic by Barr-Beck-Lurie. The composite of comonadic functors is in general not comonadic, but when the first one preserves limits we can apply Barr-Beck-Lurie to see the composite is again comonadic.

For prior work here see Gunnar Carlsson's "Derived completions in stable homotopy theory" which shows the functor is pre-comonadic, or as you say it's comonadic at every object