Locally finite p-groups with all subgroups either subnormal or nilpotent-by-Chernikov

We continue our investigation of the class X of groups G in which all subgroups are either subnormal or nilpotent-by-Chernikov. In the paper by the second author, Groups with all subgroups subnormal or nilpotent-by-Chernikov (Rend. Sem. Mat. Univ. Padova, 126 (2011), 245–253), it was shown that a locally soluble-by-finite group G in the class X is soluble if it is not nilpotent-by-Chernikov, and in any case it is soluble-by-finite. In the subsequent paper by Cutolo and Smith, Locally finite groups with all subgroups subnormal or nilpotent-by-Chernikov (Centr. Eur. J. Math., to appear), a necessary and sufficient condition was established for a locally finite group G in the class X to fail to be nilpotent-by-Chernikov, on the further assumption that all p-sections of G are nilpotent-by-Chernikov. Our concern in the present article is with p-groups in the class X, and our main result is that a Baer p-group in the class X is nilpotent-by-Chernikov.