I was wondering the same recently, and it seems to my that the answer is yes (you can get rid of reflexivity). Look at the paper of Jacques Simon : Compact sets in the spaces $L^p(0,T,B)$.
The paper claims to give sharp results in any regard and as far as I can see it only asks the spaces to be banach he gives for example Corollary 4 :
if $\{F\}$ is bounded in $L^q(0,T,X), u^\prime\in L^1(0,T,Y),$ with the usual assumption :$$X\underset{compact}{\hookrightarrow} B\underset{continous}{\hookrightarrow}Y,$$ then $\{F\}$ is relatively compact in $L^p(0,T,B)$, for $p<q$ where $X,B,Y$ are only Banach (assumption 8.1 in the paper). The corresponding result holds for $\{F\}\subset L^\infty$ and $\{F^\prime\}\subset L^r$ with $r>1$ (gives relative compactness in $\mathcal{C}(0,T,B)$).
I guess this is why it is sometimes mentionned as Aubin-Lions-Simon's lemma ...