**Edit**: The formulation of my question was incorrect, for several reasons. Here is what I hope to be the correct formulation:

Let $\mathbb{P}$ be a projective space, and $V$ a *general* linear subspace of $H^0(\mathcal{O}_{\mathbb{P}}(d))$ (that is, a general point in the corresponding Grassmannian). Then for $p<d$ the multiplication map
$$H^0(\mathcal{O}_{\mathbb{P}}(p))\otimes V\rightarrow H^0(\mathcal{O}_{\mathbb{P}}(p+d))$$
is of maximal rank, i.e. either injective or surjective.

Is this true? Known? Sasha's answer shows that it is true when $\dim V \leq \dim \Bbb{P}+1$.