A search for a narrow baryonic state in the $pK^0_S$ and $bar{p}K^0_S$ system has been performed in $ep$ collisions at HERA with the ZEUS detector using an integrated luminosity of 358 pb$^{-1}$ taken in 2003-2007. The search was performed with deep inelastic scattering events at an $ep$ centre-of-mass energy of 318 GeV for exchanged photon virtuality, $Q^2$, between 20 and 100 $rm{} GeV^{2}$. Contrary to evidence presented for such a state around 1.52 GeV in a previous ZEUS analysis using a sample of 121 pb$^{-1}$ taken in 1996-2000, no resonance peak was found in the $p(bar{p})K^0_S$ invariant-mass distribution in the range 1.45-1.7 GeV. Upper limits on the production cross section are set.