  We performed magnetization measurements in a single crystal of the
anisotropic bilayer pnictide superconductor KCa$_2$Fe$_4$As$_4$F$_2$, with
$T_c$ $\simeq$ 34 K, for $H$$\parallel$$c$-axis and $H$$\parallel$$ab$-planes.
A second magnetization peak (SMP) was observed in the isothermal $M(H)$ curves
measured below 16 K for $H$$\parallel$$ab$-planes. A peak in the temperature
variation of the critical current density, $J_{c}$($T$), at 16 K, strongly
suggests the emergence of Josephson vortices at lower temperatures, which leads
to the SMP in the sample. In addition, it is noticed that the appearance of
Josephson vortices below 16 K renders easy magnetic flux penetration. A
detailed vortex dynamics study suggests that the SMP can be explained in terms
of elastic pinning to plastic pinning crossover. Furthermore, contrary to the
common understanding, the temperature variation of the first peak field, $H_1$,
below and above 16 K, behaves non-monotonically. A highly disordered vortex
phase, governed by plastic pinning, has been observed between 17 K and 23 K,
within a field region around an extremely large first peak field. Pinning force
scaling suggests that the point defects are the dominant source of pinning for
$H$ $\parallel$$ab$-planes, whereas, for H $\parallel$$c$-axis, point defects
in addition to surface defects are at play. Such disorder contributes to the
pinning due to the variation in charge carrier mean free path,
$\delta$$l$-pinning. Moreover, the large $J_c$ observed in our study is
consistent with the literature, which advocates this material for high magnetic
field applications.
