In this paper, we evaluate, up to QCD next-to-next-to-next-to-leading order, the $B_c^{⁎}$ decay constant, which, within the nonrelativistic QCD (NRQCD) framework, is factorized as the product of the short-distance coefficient (SDC) and the long-distance matrix element. For the first time, the renormalization constant and the anomalous dimension for the NRQCD vector current composed of $c\overline{b}$, which are functions of the charm quark mass $m_c$, the bottom quark mass $m_b$ and the factorization scale ${\mu }_{\Lambda }$, are obtained analytically at $O\left({\alpha }_s^2\right)$ and $O\left({\alpha }_s^3\right)$. The SDC is calculated up to $O\left({\alpha }_s^3\right)$ in perturbative expansion. Explicitly, the $O\left({\alpha }_s^2\right)$ correction to the SDC is analytically calculated in terms of logarithmic and polylogarithmic functions of $r\equiv m_b/m_c$, and the $O\left({\alpha }_s^3\right)$ correction to the SDC is numerically calculated at a series of values of r, ranging from 2.1 to 4.0. Surprisingly, we find that the nontrivial part of the SDC at $O\left({\alpha }_s^3\right)$ can be well estimated by a linear function of r. In addition, we find that the $O\left({\alpha }_s^2\right)$ and $O\left({\alpha }_s^3\right)$ corrections to the decay constant and decay width are considerable and very significant, which indicates a very poor convergence for the perturbative expansion.