We present a low-rank method for conforming multipatch discretizations of compressible linear elasticity problems using Isogeometric Analysis. The proposed technique is a non- trivial extension of [M. Montardini, G. Sangalli, and M. Tani. A low-rank isogeometric solver based on Tucker tensors. Comput. Methods Appl. Mech. Engrg., page 116472, 2023] to multipatch geometries. We tackle the model problem using an overlapping Schwarz method, where the subdomains can be defined as unions of neigh- bouring patches. Then on each subdomain we approximate the blocks of the linear system matrix and of the right-hand side vector using Tucker matrices and Tucker vectors, re- spectively. We use the Truncated Preconditioned Conjugate Gradient as a linear solver, coupled with a suited preconditioner. The numerical experiments show the advantages of this approach in terms of memory storage. Moreover, the number of iterations is robust with respect to the relevant parameters.