  In the paper, we apply the $k_T$ factorization approach to deal with the
$B\to K$ transition form factor $F^{B\to K}_{+,0}(q^2)$ in the large recoil
regions. The B-meson wave functions $\Psi_B$ and $\bar\Psi_B$ that include the
three-particle Fock states' contributions are adopted to give a consistent PQCD
analysis of the form factor up to ${\cal O} (1/m^2_b)$. It has been found that
both the wave functions $\Psi_B$ and $\bar\Psi_B$ can give sizable
contributions to the form factor and should be kept for a better understanding
of the $B$ meson decays. Then the contributions from different twist structures
of the kaon wavefunction are discussed, including the $SU_f(3)$-breaking
effects. A sizable contribution from the twist-3 wave function $\Psi_p$ is
found, whose model dependence is discussed by taking two group of parameters
that are determined by different distribution amplitude moments obtained in the
literature. It is also shown that $F^{B\to K}_{+,0}(0)=0.30\pm0.04$ and
$[F^{B\to K}_{+,0}(0)/F^{B\to \pi}_{+,0}(0)]=1.13\pm0.02$, which are more
reasonable and consistent with the light-cone sum rule results in the large
recoil regions.
