The $kappa$-deformed exponential $ exp_kappa ( x) = exp left( { 1 over kappa } arcsinh ( kappa x ) ight) $ studied by Kaniadakis cite{K} allows to construct ``deformed" Lorentz transformations associated with the ordinary velocity boost rapidity parameter $ xi$ and which can be recast in terms of ordinary Lorentz transformations (involving the ordinary exponential) but associated with a $kappa$-deformed (modified) rapidity parameter $ xi_kappa = xi f ( kappa xi ) $ given by a $ xi$-dependent scaling of the original $ xi$ rapidity parameter. It is shown that when $both$ the $ kappa$ parameter and $ xi ightarrow infty$, and the double scaling limit $ xi { ln ( 2 kappa xi ) over kappa xi } = infty times 0 $ is finite and nonzero, it leads to a $finite$ value for the $kappa$-deformed boost rapidity parameter $ xi_{kappa = infty} = xi_{ infty} ot= infty $, such that the $kappa$-deformed velocity (in units of $ c = 1$) $ tanh ( xi_kappa ) = v_kappa < 1 $ is $less$ than the speed of light, and in turn, the Lorentz dilation factor $ gamma ( v_kappa) ot= infty $ $no$ longer $blows$ up. Consequently, there is a $lower$ bound in the length $ L' = { L over gamma ( v_k ) } ot= 0 $ due to a $finite$ Lorentz length contraction. After imposing that the lower bound $ L' $ should not be smaller than the postulated minimum Planck scale one arrives at the length-scale-dependent relation $ arccosh ( { L over L_P} ) = xi_infty > 0 $ that admits a physical interpretation analogous to the running of the physical couplings and masses with the energy scale in the Renormalization Group program in Quantum Field Theory.