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In this paper we consider the class of H\"older-Lipschitz maps on unit ball$B_X$ of a Banach space $X$, and the question we deal with is whether for any$\alpha\in (0,1)$ and $\lambda>0$ there exists a fixed-point free map $T\colonB_X\to B_X$ such that $\|Tx - Ty\|\leq \lambda\|x - y\|^\alpha$ for all $x,y\in B_X$ and $\mathrm{d}(T,B_X)=0$. We show that if $X$ has a spreadingSchauder basis then such a map can always be built, answering a question posedby the first author in \cite{Bar}. In the general case, using a recent approachof R. Medina \cite{M} concerning H\"older retractions of $(r_n)$-flat closedconvex sets, we show that for any decreasing null sequence $(r_n)\subset\mathbb{R}$, there exists a fixed-point free mapping $T$ on $B_X$ so that$\|T^nx - T^n y\|\leq r_n(\| x - y\|^\alpha +1)$ for all $x, y\in B_X$ and$n\in\mathbb{N}$. New results related to Lipschitz maps are also obtained.

Retraction methods and fixed point free maps with null minimal displacements on unit ball