One can derive the answer "Not in general" working out suitable variations of the following lemma.

**Lemma.** Let $X$ be a separable Banach space which is not isomorphic to a Hilbert space, but is continuously and injectively embedded into a Hilbert space $H$. Then $X$ contains a sequence $\{x_i\}_{i=1}^\infty$, which is linearly independent, has dense span in $X$, is an orthonormal sequence in $H$, but is not a Schauder basis in $X$ because the linear spans of finite subsets $\{x_i\}_{i=1}^n$ have indefinitely increasing projection constants in $X$.

**Proof:** By the Lindenstrauss-Tzafriri theorem [Israel J. Math. 9 (1971), 263–269] we can find a sequence $\{y_i\}_{i=1}^\infty$ in $X$ with the dense linear span satisfying the last condition. Applying the Gram-Schmidt ortonormalization, we get the desired sequence $\{x_i\}_{i=1}^\infty$.