Infinite product of reflexives Banach spaces

Question

If $X_1$ and $X_2$ are reflexive Banach spaces and $p\in[1,\infty]$, then the product $X_1\times X_2$ with the norm $\|(u,v)\|=(\|u\|^p+\|v\|^p)^{\frac1p}$ is also a reflexive Banach space (all these norms are equivalent; the $\infty$-norm is simply the supremum of the values).

Now, let $(X_i)_{i\in I}$ be a collection of reflexives Banach spaces. Given a $p\in[1,\infty]$, one can define the corresponding Cartesian products $\bigoplus^p_IX_i$ as the set of all sequences in the product whose $p$-norm is finite:

$$\bigoplus^p_IX_i=\Big\{f\in \prod_IX_i\,\Big|\,\sum_I\|f(i)\|^p \text{ is finite}\Big\}$$ endowed with the $p$-norm. These are Banach spaces, but in general $\bigoplus^p_IX_i$ and $\bigoplus^q_IX_i$ are distinct (for $p=1$ this is the coproduct, and for $p=\infty$ the product).

Question

Is these spaces reflexive? Does the answer depend on $p$?

References would be appreciated.