We present a catalog of relative proper motions for 368,787 stars in the 30 Doradus region of the Large Magellanic Cloud (LMC), based on a dedicated two-epoch survey with the Hubble Space Telescope (HST) and supplemented with proper motions from our pilot archival study. We demonstrate that a relatively short epoch difference of 3 years is sufficient to reach a $sim$0.1 mas yr$^{-1}$ level of precision or better. A number of stars have relative proper motions exceeding a 3-sigma error threshold, representing a mixture of Milky Way denizens and 17 potential LMC runaway stars. Based upon 183 VFTS OB-stars with the best proper motions, we conclude that none of them move faster than $sim$0.3 mas yr$^{-1}$ in each coordinate -- equivalent to $sim$70 km s$^{-1}$. Among the remaining 351 VFTS stars with less accurate proper motions, only one candidate OB runaway can be identified. We rule out any OB star in our sample moving at a tangential velocity exceeding $sim$120 km s$^{-1}$. The most significant result of this study is finding 10 stars over wide range of masses, which appear to be ejected from the massive star cluster R136 in the tangential plane to angular distances from $35^{primeprime}$ out to $407^{primeprime}$, equivalent to 8-98 pc. The tangential velocities of these runaways appear to be correlated with apparent magnitude, indicating a possible dependence on the stellar mass.
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