We consider non-Abelian strings in N=2 supersymmetric QCD with the U(N) gauge group and N_f=N quark flavors deformed by a mass term for the adjoint matter. This deformation breaks N=2 supersymmetry down to N=1. Dynamics of orientational zero modes on the string world sheet are described then by CP(N-1) model with N=(0,2) supersymmetry. We study the string of a finite length L assuming compactification on a cylinder (periodic boundary conditions). The world-sheet theory is solved in the large-N

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... pproximation. We find a rich phase structure in the (L, u) plane where u is a deformation parameter. At large L and intermediate u we find a phase with broken Z_2N symmetry, N vacua and a mass gap. At large values of L and u still larger we have the Z_2N-symmetric phase with a single vacuum and massless fermions. In both phases N=(0,2) supersymmetry is spontaneously broken. We also observe a phase with broken SU(N) symmetry at small L. In the latter phase the mass gap vanishes and the vacuum energy is zero in the leading 1/N approximation. However, we expect that 1/N corrections will break N=(0,2) supersymmetry. We also discuss how this rich phase structure matches the N=(2,2) limit in which the world-sheet theory has a single phase with the mass gap independent of L.