  Let $X$ be a metric space with bounded geometry, $p\in\{0\} \cup [1,\infty]$,
and let $E$ be a Banach space. The main result of this paper is that either if
$X$ has Yu's Property A and $p\in(1,\infty)$, or without any condition on $X$
when $p\in\{0,1,\infty\}$, then quasi-local operators on $\ell^p(X,E)$ belong
to (the appropriate variant of) Roe algebra of $X$. This generalises the
existing results of this type by Lange and Rabinovich, Engel, Tikuisis and the
first author, and Li, Wang and the second author. As consequences, we obtain
that uniform $\ell^p$-Roe algebras (of spaces with Property A) are closed under
taking inverses, and another condition characterising Property A, akin to
operator norm localisation for quasi-local operators.
