A real Banach space \(X\) is said to have the Mazur-Ulam property if the following holds: every surjective isometry from the unit sphere \(S_X\) of \(X\) onto the unit sphere \(S_Y\) of another Banach space \(Y\) can be extended to a linear isometry between \(X\) and \(Y\).\par The main result of the paper reads as follows. Suppose that \(X\) is a strictly convex Banach space with \(\mathrm{dim}(X)\geq 2\) such that the smooth points of \(S_X\) are norm dense in \(S_X\). Assume further that \(K\) is a totally disconnected, locally compact Hausdorff space with \(|K|\geq 2\). Then \(C_0(K,X)\) (the space of all continuous functions from \(K\) to \(X\) that vanish at infinity) has the Mazur-Ulam property.