The work analyzes the effect of a magnetic field $B$ directed along the $c$ axis ($B \parallel c$) up to 9~T on the resistivity $\rho(T)$, fluctuation conductivity (FLC) $\sigma'(T)$ and pseudogap $\Delta^*(T)$ in thin films of YBa$_2$Cu$_3$O$_{7-\delta}$ with a critical temperature of the superconducting transition $T_c = 88.8$~K. In contrast to previous work, where the magnetic field was directed along the $ab$ plane ($B \parallel ab$), the influence of the field on the sample is stronger due to the contribution of both spin--orbit and Zeeman effects. It was found that the BEC--BCS transition temperature, $T_{\mathrm{pair}}$, which corresponds to the maximum of the $\Delta^*(T)$ dependence, shifts to the region of lower temperatures with increasing $B$, and the maximum value of $\Delta^*(T_{\mathrm{pair}})$ decreases in fields $B > 5$~T. It was found that with increasing field, the low-temperature maximum near $T_0$ is smeared and disappears at $B > 1$~T. In addition, above the Ginzburg temperature $T_G$, for $B > 1$~T, a minimum appears on $\Delta^*(T)$ at $T_{\min}$, which becomes very pronounced with a subsequent increase in $B$. As a result, the overall value of $\Delta^*(T_G)$ decreases noticeably, most likely due to the pair-breaking effect. At the same time, $\Delta T_{\mathrm{fl}}$ and $\xi_c(0)$ increase sharply by approximately 3 times with increasing $B$ above 1~T. Our results confirm the possibility of the formation of a vortex state in YBa$_2$Cu$_3$O$_{7-\delta}$ by a magnetic field and its evolution with increasing $B$.