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We propose commuting set of matrix-valued difference operators in terms oftrigonometric ${\rm GL}(N|M)$-valued $R$-matrices providing quantumsupersymmetric (and possibly anisotropic) spin Ruijsenaars-Macdonald operators.Two types of trigonometric supersymmetric $R$-matrices are used. The first isthe one related to the affine quantized algebra ${\hat{\mathcal U}}_q({\rmgl}(N|M))$. The second is a graded version of the standard $\mathbbZ_n$-invariant $A_{n-1}$ type $R$-matrix. We show that being properlynormalized the latter graded $R$-matrix satisfies the associative Yang-Baxterequation. Next, we proceed to construction of long-range spin chains using thePolychronakos freezing trick. As a result we obtain a new family of spinchains, which extend the ${\rm gl}(N|M)$-invariant Haldane-Shastry spin chainto q-deformed case with possible presence of anisotropy.

Supersymmetric generalization of q-deformed long-range spin chains of  Haldane-Shastry type and trigonometric GL(N|M) solution of associative  Yang-Baxter equation