A search for the doubly heavy $\\mathit{\\Xi}_{bc}^{0}$ baryon using its decay to the $D^0pK^-$ final state is performed using proton-proton collision data at a centre-of-mass energy of 13 TeV collected by the LHCb experiment between 2016 and 2018, corresponding to an integrated luminosity of 5.4 $\\mathrm{fb}^{-1}$. No significant signal is found in the invariant mass range from 6.7 to 7.2 $\\mathrm{GeV}/c^2$. Upper limits are set at $95\\%$ credibility level on the ratio of the $\\mathit{\\Xi}_{bc}^{0}$ production cross-section times its branching fraction to $D^0pK^-$ relative to that of the $\\mathit{\\Lambda}_{b}^{0} \\to D^0pK^-$ decay. The limits are set as a function of the $\\mathit{\\Xi}_{bc}^{0}$ mass and lifetime hypotheses, in the rapidity range from 2.0 to 4.5 and in the transverse momentum region from 5 to 25 $\\mathrm{GeV}/c$. Upper limits range from $1.7\\times10^{-2}$ to $3.0\\times10^{-1}$ for the considered $\\mathit{\\Xi}_{bc}^{0}$ mass and lifetime hypotheses.
