# Matrix of compact operator on Banach space of the form $\ell^p(Z,\ell^p)$

For a countable subset $$Z$$ of a metric space $$X$$, consider the Banach space $$\ell^p(Z,\ell^p)$$ for $$p\in[1,\infty)$$. The space of continuous functions $$C_0(X)$$ acts on $$\ell^p(Z,\ell^p)$$ by multiplication (restricted to $$Z$$). Let $$T$$ be a compact operator on $$\ell^p(Z,\ell^p)$$, regarded as a $$Z$$-by-$$Z$$ matrix of bounded operators on $$\ell^p$$. Is it necessarily the case that for any bounded subset $$B\subset X$$, the set of $$(x,y)\in B\times B$$ such that $$T_{xy}\neq 0$$ is finite?

(The background behind this question is that I am comparing two definitions:

1. $$fT$$ and $$Tf$$ are compact operators on $$\ell^p(Z,\ell^p)$$ for all $$f\in C_0(X)$$.

2. Each $$T_{xy}$$ is a compact operator on $$\ell^p$$, and for each bounded subset $$B\subseteq X$$, the set of $$(x,y)\in B\times B$$ such that $$T_{xy}\neq 0$$ is finite.)