Let $S=(0,T)$ for a $T>0$ and let $B_0,\ B_1,\ B_2$ be Banach spaces, such that $B_0$ is compactly embedded in $B_1$, which is in turn continuously embedded in $B_2$.
Suppose we have a sequence $f_n\in W:=\{u\in L^2(S;B_0) \ : \ \partial_t u\in L^2(S;B_2)\}$, $n\in\mathbb{N}$, such that $\sup_{n\in\mathbb{N}}\|f_n\|_{W}<\infty$. Does this imply the existence of subsequence $f_{n_k}$ and a function $f\in L^2(S;B_1)$ such that $f_{n_k}$ converges strongly in $L^2(S;B_1)$ to $f$ for ${k\to\infty}$.
Since the compactness of the embedding $W\hookrightarrow L^2(S;B_1)$ holds by virtue of the compactness lemma of Lions/Aubin, it seems to me that this question can be reduced to the question "Is $W$ reflexive?". Is this correct? And if so, which assumptions on $B_0$, $B_2$ ensure the reflexivity of $W$? Is reflexivity of $B_0$, $B_2$ necessary and/or sufficient?
If there is an interpolation-like inequality to estimate the norms $$ \|v\|_{B_1}\le c \|v\|_{B_0}^\theta\|v\|_{B_2}^{1-\theta}\quad \forall v\in B_0 $$ with $c>0$ and $\theta\in(0,1)$, then the answer to your question is yes. The result can be found in
Amann: Compact embeddings of vector-valued Sobolev and Besov spaces http://web.math.pmf.unizg.hr/glasnik/vol_35/no1_10.html
This result does not need any reflexivity.