Mechanical property assessment by elastographic ultrasound methods, including those that exploit acoustic radiation force (ARF), relies on accurate estimation of tissue displacement. Several displacement estimators have been developed, but their relevance to tracking ARF-induced displacements smaller than one micrometer is limited by estimation variance. To minimize displacement estimation variance, a new blind source separation (BSS)-based approach that exploits the spatial distribution of displacements is presented. This new approach applies principal component analysis (PCA) in three-dimensional (3D) kernels to derive dominant eigenvectors, from which displacements are estimated. We call this new approach the '3DK-BSS' estimator. The 3DK-BSS estimator is evaluated in terms of contrast-to-noise ratio (CNR) achieved in ARFI peak displacement (PD) images of a stiff (80 kPa) and a soft (8 kPa) spherical inclusion in a commercial phantom with a 25 kPA background. The CNR values achieved by 3DK-BSS were compared to those produced by normalized cross-correlation (NCC), Bayesian regularization, and BSS implemented using a two-dimensional (2D) kernel at ARF power levels of 5 to 45% of the full system power. For all power levels, and for both stiff and soft inclusions, 3DK-BSS inclusion CNR was higher than any other examined displacement estimator. For example, 3DK-BSS stiff inclusion CNR was 16.3, 11.8, and 4.2 times higher than the CNRs achieved by NCC, Bayesian regularization, and 2D BSS, respectively at 5% power level. CNR improvement by 3DK-BSS was largest at the lowest (5%) ARF power level, where displacement in the stiff phantom was measured by 3DK-BSS as 250 nm. These results suggest that, by accurately measuring sub-micrometer displacements, the 3DK-BSS displacement estimator could enable deeper ARF-based mechanical property assessments, finer mechanical resolution in stiff tissues, and/or lower ARF power requirements to expand the diagnostic relevance of ARF-based mechanical property assessments.