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We present the measurement of the energy dependence of the boron flux in cosmic rays and its ratio to the carbon flux \textcolor{black}{in an energy interval from 8.4 GeV$/n$ to 3.8 TeV$/n$} based on the data collected by the CALorimetric Electron Telescope (CALET) during $\sim 6.4$ years of operation on the International Space Station. An update of the energy spectrum of carbon is also presented with an increase in statistics over our previous measurement. The observed boron flux shows a spectral hardening at the same transition energy $E_0 \sim 200$ GeV$/n$ of the C spectrum, though B and C fluxes have different energy dependences. The spectral index of the B spectrum is found to be $γ= -3.047\pm0.024$ in the interval $25 < E < 200$ GeV$/n$. The B spectrum hardens by $Δγ_B=0.25\pm0.12$, while the best fit value for the spectral variation of C is $Δγ_C=0.19\pm0.03$. The B/C flux ratio is compatible with a hardening of $0.09\pm0.05$, though a single power-law energy dependence cannot be ruled out given the current statistical uncertainties. A break in the B/C ratio energy dependence would support the recent AMS-02 observations that secondary cosmic rays exhibit a stronger hardening than primary ones. We also perform a fit to the B/C ratio with a leaky-box model of the cosmic-ray propagation in the Galaxy in order to probe a possible residual value $λ_0$ of the mean escape path length $λ$ at high energy. We find that our B/C data are compatible with a non-zero value of $λ_0$, which can be interpreted as the column density of matter that cosmic rays cross within the acceleration region.