Recently a one-parameter extension of the covariant Heisenberg algebra with the extension parameter l (l is a non-negative constant parameter which has a dimension of [momentum] −1 ) in a (D + 1)dimensional Minkowski space-time has been presented [G. P. de Brito, P. I. C. Caneda, Y. M. P. Gomes, J. T. Guaitolini Junior and V. Nikoofard, Effective models of quantum gravity induced by Planck scale modifications in the covariant quantum algebra, Adv. High Energy Phys. 2017 (2017) 4768341]. The Abelian Proca model is reformulated from the viewpoint of the above one-parameter extension of the covariant Heisenberg algebra. It is shown that the free space solutions of the above modified Proca model satisfy the modified dispersion relation p 2 1+ Λ 2 2 2 p 2 2 = m 2 c 2 where Λ = l is the characteristic length scale in our model. This modified dispersion relation describes two massive vector particles with the effective masses M ± (Λ) = 2m 1∓ 1−2( mcΛ ) 2 . Numerical estimations show that the maximum value of Λ in a four-dimensional space-time is near to the electroweak length scale, i.e., Λ max ∼ l electroweak ∼ 10 −18 m. We show that in the infrared/large-distance domain the modified Proca model behaves like an Abelian massive Lee-Wick model which has been presented by Accioly and his co-workers [A. Accioly, J. Helayel-Neto, G. Correia, G. Brito, J. de Almeida and W. Herdy, Interparticle potential energy for D-dimensional electromagnetic models from the corresponding scalar ones, Phys. Rev. D 93 (2016) 105042].