  We continue our study of 'no-dimension' analogues of basic theorems in
combinatorial and convex geometry in Banach spaces. We generalize some results
of the paper \cite{adiprasito2019theorems} and prove no-dimension versions of
colorful Tverberg's theorem, selection lemma and the weak $\epsilon$-net
theorem in Banach spaces of type $p > 1.$ To prove this results we use the
original ideas of \cite{adiprasito2019theorems} for the Euclidean case and our
slightly modified version of the celebrated Maurey lemma.
