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The known quantum error-correcting codes are typically built on approximative open-quantum-system models such as Born--Markov master equations. However, it is an open question how such codes perform in actual physical systems that, to some extent, necessarily exhibit phenomena beyond the limits of these models. To this end, we employ numerically exact open-quantum-system dynamics to analyze the performance of a five-qubit error correction code where each qubit is coupled to its own bath. We first focus on the performance of a single error correction cycle covering time scales beyond that of Born--Markov models. Namely, we observe distinct power law behavior of the channel infidelity $\propto t^{2a}$: $a\lesssim 2$ in the ultrashort times $t<3/ω_{\rm c}$ and $a\approx 1/2$ in the short-time range $3/ω_{\rm c}<t<30/ω_{\rm c}$, where $ω_{\rm c}$ is the cutoff angular frequency of the bath. Importantly, the five-qubit quantum-error correction code suppresses all single errors, including those arising from the ultrashort and short-time evolution, which are peculiar to the exact evolution. Interestingly, we demonstrate the breaking points of the five-qubit error correction code and the Born--Markov models for repeated error correction when the repetition rate exceeds $2π/ω$ or the coupling strength $κ\gtrsim 0.1 ω$, where $ω$ is the angular frequency of the qubit. Our results pave the way for applying numerically exact open-quantum-system models for the studies of QECs beyond simple error models.