Let $A$ be a unital C*-algebra and $\mathcal{E}$ a Hilbert A-module. If $\{m_j\}$ is a finite Parseval frame for $\mathcal{E}$ and $\{m_j\}$ is orthogonal, is it actually a base? I suspect that's not the case because $\langle m_i,m_i \rangle b=0$ doesn't necessarily imply that $b=0$.
Note: $\{m_j\}$ is a Parseval frame if for all $m\in \mathcal{E}$ we have that $$m=\sum_j m_j\langle m_j ,m \rangle$$
I suppose, by "base" you mean a linearly independent set which spans $\mathcal E$ as a complex vector space.
You are right. Let $A$ be any unital C*-algebra. Consider $A$ as the standard right Hilbert $A$-module. Then $\{1_A\}$ is a frame but clearly not a base, if $A$ has complex dimension greater than one.