Ultra-faint galaxies are hosted by small dark matter halos with shallow gravitational potential wells, hence their star formation activity is more sensitive to feedback effects. The shape of the faint-end of the high-$z$ galaxy luminosity function (LF) contains important information on star formation and its interaction with the reionization process during the Epoch of Reionization (EoR). High-$z$ galaxies with $M_{rm UV}gtrsim-17$ have only recently become accessible thanks to the Frontier Fields (FFs) survey combining deep {it HST} imaging and the gravitational lensing effect. In this paper we investigate the faint-end of the LF at redshift $>$5 using the data of FFs clusters Abell 2744 (A2744), MACSJ0416.1-2403 (M0416), MACSJ0717.5+3745 (M0717) and MACSJ1149.5+2223 (M1149). We analyze both an empirical and a physically-motivated LF model to obtain constraints on a possible turn-over of LF at faint magnitudes. In the empirical model the LF drops fast when the absolute UV magnitude $M_{rm UV}$ is much larger than a turn-over absolute UV magnitude $M_{rm UV}^{rm T}$. We obtain $M_{rm UV}^{rm T}gtrsim-14.6 $ (15.2) at 1 (2) $sigma$ confidence level (C.L.) for $zsim6$. In the physically-motivated analytical model, star formation in halos with circular velocity below $v_c^*$ is fully quenched if these halos are located in ionized regions. Using updated lensing models and new additional FFs data, we re-analyze previous constraints on $v_c^*$ and $f_{rm esc}$ presented by Castellano et al. 2016a (C16a) using a smaller dataset. We obtain new constraints 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 so far from the analyzed FFs data. Forthcoming {it JWST} observations will be key to tight these constraints further.
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