This is  [ 1 , Lemma II.9, p.15 ] .

You can assume that \(S\) is affine and, then, choose an embedding \(j:S\hookrightarrow \mathbb {A}^n\) for some \(n\). We get a rational map \(j\circ f^{-1}:S'\dashrightarrow \mathbb {A}^n\), which is then given by rational functions \(g_1,\ldots , g_n\) on \(S'\). By assumption, one of them is not defined at \(p\), and, without loss of generality, we can assume it is \(g_1\). So \(g_1\not\in \mathcal{O}_{S',p}\). Since \(S'\) is smooth, \(\mathcal{O}_{S',p}\) is factorial. So we can write \(g_1 = u/v\) with \(u,v\in \mathcal{O}_{S',p}\) coprime and \(v\not\in \mathfrak {m}_{p}\). This means that \(f^*(u/v) = x_1\) or that \(f^* u = x_1 f^* v\), with \(x_1\) the first coordinate on \(S\subseteq \mathbb {A}^n\). But then, if \(D = V(f^*v) \subseteq S\), we get that \(f^* u = f^* v = 0\) on \(D\). So that \(D = f^{-1} (Z)\), where \(Z = V((u,v))\subseteq S'\). But then \(Z\) is finite, so we can assue that it is just the point \(p\), and we get what we want.