I need to construct such a function but the closest I have come to is to take $f(t) = e^{-|t|}, t\in\Bbb{R^d}$. But its Fourier transform is not compactly supported as is $\hat{f}(x) = \frac{2}{1+x^2}$.
this question is closely related, but the requirement of the function is reversed.
Take any nontrivial, nonnegative, symmetric function $g \in C_c^\infty (\Bbb{R}^d)$. If we let $h := \mathcal{F}^{-1}(g)$, then $h$ is real-valued (why?) and $$ h(0) = \int g(x) \, dx > 0, $$ since $g \geq 0$ and $g \not \equiv 0$.
By continuity of $h$ and by rescaling (i.e., replace $g$ by $Cg$ for some large $C>1$), we get $h \geq 1$ on $B_{2\delta} (0)$ for some $\delta > 0$.
Now, finally take $f : x \mapsto h(\delta \cdot x)$. It is not hard to see $f \geq 1$ on $B_1 (0)$ and $$ \widehat{f} = \delta^{-d} \cdot \widehat{h}(\cdot / \delta) \in C_c^\infty, $$ as desired.