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Ultra-faint galaxies are hosted by small dark matter halos with shallowgravitational potential wells, hence their star formation activity is moresensitive to feedback effects. The shape of the faint-end of the high-$z$galaxy luminosity function (LF) contains important information on starformation and its interaction with the reionization process during the Epoch ofReionization (EoR). High-$z$ galaxies with $M_{\rm UV}\gtrsim-17$ have onlyrecently become accessible thanks to the Frontier Fields (FFs) survey combiningdeep {\it HST} imaging and the gravitational lensing effect. In this paper weinvestigate the faint-end of the LF at redshift $>$5 using the data of FFsclusters Abell 2744 (A2744), MACSJ0416.1-2403 (M0416), MACSJ0717.5+3745 (M0717)and MACSJ1149.5+2223 (M1149). We analyze both an empirical and aphysically-motivated LF model to obtain constraints on a possible turn-over ofLF at faint magnitudes. In the empirical model the LF drops fast when theabsolute UV magnitude $M_{\rm UV}$ is much larger than a turn-over absolute UVmagnitude $M_{\rm UV}^{\rm T}$. We obtain $M_{\rm UV}^{\rm T}\gtrsim-14.6 $(15.2) at 1 (2) $\sigma$ confidence level (C.L.) for $z\sim6$. In thephysically-motivated analytical model, star formation in halos with circularvelocity below $v_c^*$ is fully quenched if these halos are located in ionizedregions. Using updated lensing models and new additional FFs data, were-analyze previous constraints on $v_c^*$ and $f_{\rm esc}$ presented byCastellano et al. 2016a (C16a) using a smaller dataset. We obtain newconstraints on $v_c^*\lesssim 59$ km s$^{-1}$ and $f_{\rm esc}\lesssim 56\%$(both at 2$\sigma$ C.L.) and conclude that there is no turn-over detected sofar from the analyzed FFs data. Forthcoming {\it JWST} observations will be keyto tight these constraints further.

On the faint-end of the galaxy luminosity function in the Epoch of Reionization: updated constraints from the HST Frontier Fields