Let $X$ be a Banach space. It is well-known that if $(x_n)_{n \geq 1}$ is a boundedly complete Schauder basis of $X$, then $X$ has the Radon-Nikodým property (indeed, $X$ is separable and is canonically isomorphic to $H^*$, where $H$ is the closed linear span of the coordinate functionals associated to $(x_n)_{n \geq 1}$).
Now assume that $X$ has an uncountable (unconditional) boundedly complete basis $(x_i)_{i \in I}$. By an unconditional basis I mean a family $(x_i)_{i \in I}$ with the property that for every $x \in X$, there exists a unique family of scalaras $(\alpha_i)_{i \in I}$ such that $x = \sum_{i \in I} \alpha_i x_i$, and the sum is meant in the sense that for every $\varepsilon > 0$ there exists a finite subset $F$ of $I$ such that $$\left\| x - \sum_{i \in G} \alpha_i x_i\right\| < \varepsilon$$ whenever $G$ is a finite subset of $I$ that contains $F$.
Under such conditions, does $X$ also have the Radon-Nikodým property?