  In this paper, (gradient) almost Ricci solitons on Finsler measure spaces
$(M, F, m)$ are introduced and investigated. We prove that $(M, F, m)$ is a
gradient almost Ricci soliton if and only if the infinity-Ricci curvature
Ric$_\infty$ is a scalar function on $M$ when $M$ is compact. Moreover, we give
an equivalent characterization of (gradient) almost Ricci solitons for Randers
metrics $F=\alpha+\beta$, which implies that every Randers (gradient) almost
Ricci soliton is of isotropic S$_{BH}$-curvature. Based on this and the
navigation technique, we further classify Randers almost Ricci solitons (resp.
gradient almost Ricci solitons) up to classifications of Randers Einstein
metrics $F$ (resp. Riemannian gradient almost Ricci solitons) and the
homothetic vector fields of $F$ (resp. solutions of the equation which the
weight function $f$ of $m$ satisfies) when $F$ has isotropic
S$_{BH}$-curvature. As applications, we obtain some rigidity results for
compact Randers (gradient) Ricci solitons and construct several Randers
gradient Ricci solitons, which are the first nontrivial examples of gradient
Ricci solitons in Finsler geometry.
