In this paper, we prove the following nonlocal Kirchhoff problem of the type
has a Nehari‐type ground state solution when
and
are periodic on
and
has critical exponential growth in the sense of Trudinger–Moser inequality on
. We develop some new approaches to estimate precisely the minimax level of the energy functional and to recover the compactness of Cerami sequences of the associated Euler–Lagrange functional. With this in hand, we can overcome difficulties arising from the appearance of the nonlocal term
and the nonlinearity which is of critical growth of Trudinger–Moser type in whole Euclidean space
.