We consider the solution of large linear systems with multiple right-hand sides using a block GMRES approach. We introduce a new algorithm that effectively handles the situation of almost rank deficient block generated by the block Arnoldi procedure and that enables the recycling of spectral information at restart. The first feature is inherited from an algorithm introduced by Robbé and Sadkane~[M. Robbé and M. Sadkane. Exact and inexact breakdowns in the block GMRES method. Linear Algebra and its Applications, 419: 265-285, 2006.], while the second one is obtained by extending the deflated restarting strategy proposed by Morgan~[R. B. Morgan. Restarted block GMRES with deflation of eigenvalues. Applied Numerical Mathematics, 54(2): 222-236, 2005.]. Through numerical experiments, we show that the new algorithm combines efficiently the attractive numerical features of its two parents that it outperforms.