We present a catalog of high-precision proper motions in the Orion Nebula Cluster (ONC), based on Treasury Program observations with the Hubble Space Telescopes (HST) ACS/WFC camera. Our catalog contains 2,454 objects in the magnitude range of $14.2<m_{rm F775W}<24.7$, thus probing the stellar masses of the ONC from $sim$0.4 $M_odot$ down to $sim$0.02 $M_odot$ over an area of $sim$550 arcmin$^2$. We provide a number of internal velocity dispersion estimates for the ONC that indicate a weak dependence on the stellar location and mass. There is good agreement with the published velocity dispersion estimates, although nearly all of them (including ours at $sigma_{v,x}=0.94$ and $sigma_{v,y}=1.25$ mas yr$^{-1}$) might be biased by the overlapping young stellar populations of Orion A. We identified 4 new ONC candidate runaways based on HST and the Gaia DR2 data, all with masses less than $sim$1 $M_odot$. The total census of known candidate runaway sources is 10 -- one of the largest samples ever found in any Milky Way open star cluster. Surprisingly, none of them has the tangential velocity exceeding 20 km s$^{-1}$. If most of them indeed originated in the ONC, it may compel re-examination of dynamical processes in very young star clusters. It appears that the mass function of the ONC is not significantly affected by the lost runaways.
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