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

We pursue further our investigation, begun in [H.~Smith, Groups with all subgroups subnormal or nilpotent-by-{C}hernikov, emph{Rend. Sem. Mat. Univ. Padova} 126 (2011), 245--253] and continued in [G.~Cutolo and H.~Smith, Locally finite groups with all subgroups subnormal or nilpotent-by-{C}hernikov. emph{Centr. Eur. J. Math.} (to appear)] of groups $G$ in which all subgroups are either subnormal or nilpotent-by-Chernikov. Denoting by $mathfrak{X}$ the class of all such groups, our concern here is with locally finite p-groups in the class $mathfrak{X}$, where $p$ is a prime, while an earlier article provided a reasonable classification of locally finite $mathfrak{X}$nb-groups in which all of the p-sections are nilpotent-by-Chernikov. Our main result is that if $G$ is a Baer p-group in $mathfrak{X}$ then $G$ is nilpotent-by-Chernikov .