We propose a new class of R-estimators for semiparametric VARMA models in which the innovation density plays the role of the nuisance parameter. Our estimators are based on the novel concepts of multivariate center-outward ranks and signs. We show that these concepts, combined with Le Cam's asymptotic theory of statistical experiments, yield a class of semiparametric estimation procedures, which are efficient (at a given reference density), root-$n$ consistent, and asymptotically normal under a broad class of (possibly non elliptical) actual innovation densities. No kernel density estimation is required to implement our procedures. A Monte Carlo comparative study of our R-estimators and other routinely-applied competitors demonstrates the benefits of the novel methodology, in large and small sample. Proofs, computational aspects, and further numerical results are available in the supplementary material.