  We study $\ell^r$-valued extensions of linear operators defined on Lebesgue
spaces with variable exponent. Under some natural (and usual) conditions on the
exponents, we characterize $1\leq r\leq \infty$ such that every bounded linear
operator $T\colon L^{q(\cdot)}(\Omega_2, \mu)\to L^{p(\cdot)}(\Omega_1, \nu)$
has a bounded $\ell^r$-valued extension. We consider both non-atomic measures
and measures with atoms and show the differences that can arise. We present
some applications of our results to weighted norm inequalities of linear
operators and vector-valued extensions of fractional operators with rough
kernel.
