<p>The aim of the present paper is to propose and study a dissipative variant of symplectic billiards within planar strictly convex domains. The associated billiard map is dissipative, thus it admits a compact invariant set, the so-called Birkhoff attractor. Its complexity depends on the rate of dissipation as well as on the geometry of the billiard table. We prove that (<i>a</i>) for strong dissipation, the Birkhoff attractor is a normally contracted graph over the zero section; (<i>b</i>) for mild dissipation, the Birkhoff attractor within a centrally symmetric domain is an indecomposable continuum whose restricted dynamics has positive topological entropy. We compare these results with those for dissipative Birkhoff billiards, studied in [<CitationRef CitationID="CR15">15</CitationRef>].</p>