  We investigate the ground-state phase diagram of a binary mixture of
Bose-Einstein condensates (BECs) with competing interspecies $s$- and $p$-wave
interactions. Exploiting a pseudopotential model for the $l=1$ partial wave, we
derive an extended Gross-Pitaevskii (GP) equation for the BEC mixture that
incorporates both $s$- and $p$-wave interactions. Based on it, we study the
miscible-immiscible transition of a binary BEC mixture in the presence of
interspecies $p$-wave interaction, by combining numerical solution of the GP
equation and Gaussian variational analysis. Our study uncovers a dual effect --
either enhance or reduce miscibility -- of positive interspecies $p$-wave
interaction, which can be precisely controlled by adjusting relevant
experimental parameters. By complete characterizing the miscibility phase
diagram, we establish a promising avenue towards experimental control of the
miscibility of binary BEC mixtures via high partial-wave interactions.
