Depth notions in regression have been systematically proposed and examined in Zuo (2018). One of the prominent advantages of notion of depth is that it can be directly utilized to introduce median-type deepest estimating functionals (or estimators in empirical distribution case) for location or regression parameters in a multi-dimensional setting. Regression depth shares the advantage. Depth induced deepest estimating functionals are expected to inherit desirable and inherent robustness properties ( e.g. bounded maximum bias and influence function and high breakdown point) as their univariate location counterpart does. Investigating and verifying the robustness of the deepest projection estimating functional (in terms of maximum bias, asymptotic and finite sample breakdown point, and influence function) is the major goal of this article. It turns out that the deepest projection estimating functional possesses a bounded influence function and the best possible asymptotic breakdown point as well as the best finite sample breakdown point with robust choice of its univariate regression and scale component.