We consider the problem of nonasymptotic mean estimation for heavy-tailed random variables taking values in a smooth Banach space. In particular, we extend and improve a simple truncation-based mean estimator by Catoni and Giulini. We show that this estimator enjoys finite-sample, high probability deviation bounds that hold uniformly over the class of distributions with a $p$th central moment bounded above by a known constant $v > 0$ for $p \in (1, 2]$. Our main contributions exploit connections between truncation-based mean estimation and the concentration of martingales with bounded increments in smooth Banach spaces. We prove two types of time-uniform bounds on the distance between the estimator and unknown mean: line-crossing inequalities, which can be optimized for a fixed sample size $n$, and iterated logarithm inequalities, which match the tightness of line-crossing inequalities at all points in time up to a doubly logarithmic factor in $n$. Our results do not depend on the dimension of the Banach space, hold under martingale dependence, and all employed constants are known and small.

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