Is the Local Average of a Continuous Multivariable Function Differentiable?

Question

Suppose we have a continuous $f:\mathbb{R}\to\mathbb{R}$. It is an immediate corollary of the Leibniz integral rule that $f^*:x\mapsto\int_{x-\frac{1}{2}}^{x+\frac{1}{2}}f(t)\ dt$, the "local" average of $f$ on the interval of length one centred at $x$, is a continuously differentiable function in $x$. I was wondering how far we can get extending this result into higher dimensions. Specifically, let $f:\mathbb{R}^n\to\mathbb{R}^m$ be continuous. For $x\in\mathbb{R}^n$, let $B_r(x)$ be the closed ball of radius $r$ around $x$. Then, is the function $f^*:x\to\int_{B_{\frac{1}{2}}(x)}f(t)\ dt$, where the integral is the standard integral for vector-valued functions, a continuously differentiable function? Could we say, integrate over a square instead of a circle?