The answer is no. Indeed, consider the case when $n=2$ and $$f(x,y)=\max(| x|,2 | y|)$$ for all $(x,y)\in\mathbb R^2$.
Then for $(x,y)$ near $(\pm1,0)$, we have $f(x,y)=|x|$ and hence $\partial f(\pm1,0)\subseteq B_1(0)$. On the other hand, $f(0,y)=2|y|$, so that $\partial f(0,0)\not\subseteq B_1(0)$.