Perturbation of a set of vectors which spans the whole space

Question

I came across the following question when I was reading some linear algebra related stuff: Suppose we have a set of vectors $\{\vec{u}_1,\dots,\vec{u}_m\}$ in $\mathbb{R}^n$ such that $\text{span}\{\vec{u}_1,\dots,\vec{u}_m\}=\mathbb{R}^n$. Now let $\vec{p}_1,\dots,\vec{p}_m\in\mathbb{R}^d$ be such that $$\max\{\|\vec{p}_i\|:i=1,\dots,m\}<\delta,\quad (*)$$ where $\delta>0$. Then is it possible to choose $\delta$ small enough such that the set $\{\vec{u}_1+\vec{p}_1,\dots,\vec{u}_m+\vec{p}_m\}$ still spans the whole space $\mathbb{R}^n$ whenever $(*)$ holds? I think the answer is positive. But it seems that this is not a very trivial question. Any suggestions are welcomed.