By Lemma \ref{lemma-neron-functorial} we may replace $A$ by a localization at an element not in $\mathfrak p$; we will use this without further mention. Write $\kappa = R/\pi R$. Since smoothness is stable under base change (Algebra, Lemma \ref{algebra-lemma-base-change-smooth}) we see that $A/\pi A$ is smooth over $\kappa$ at $\mathfrak p$. Hence $(A/\pi A)_\mathfrak p$ is a regular local ring (Algebra, Lemma \ref{algebra-lemma-characterize-smooth-over-field}). Choose $g_1, \ldots, g_c \in \mathfrak p$ which map to a regular system of parameters in $(A/\pi A)_\mathfrak p$. Then we see that $\mathfrak p = (\pi, g_1, \ldots, g_c)$ after possibly replacing $A$ by a localization. Note that $\pi, g_1, \ldots, g_c$ is a regular sequence in $A_\mathfrak p$ (first $\pi$ is a nonzerodivisor and then Algebra, Lemma \ref{algebra-lemma-regular-ring-CM} for the rest of the sequence). After replacing $A$ by a localization we may assume that $\pi, g_1, \ldots, g_c$ is a regular sequence in $A$ (Algebra, Lemma \ref{algebra-lemma-regular-sequence-in-neighbourhood}). It follows that $$ A' = A[y_1, \ldots, y_c]/(\pi y_1 - g_1, \ldots, \pi y_c - g_c) = A[y_1, \ldots, y_c]/I $$ by More on Algebra, Lemma \ref{more-algebra-lemma-blowup-regular-sequence}. In the following we will use the definition of smoothness using the naive cotangent complex (Algebra, Definition \ref{algebra-definition-smooth}) and the criterion of Algebra, Lemma \ref{algebra-lemma-smooth-at-point} without further mention. The exact sequence of Algebra, Lemma \ref{algebra-lemma-exact-sequence-NL} for $R \to A[y_1, \ldots, y_c] \to A'$ looks like this $$ 0 \to H_1(\NL_{A'/R}) \to I/I^2 \to \Omega_{A/R} \otimes_A A' \oplus \bigoplus\nolimits_{i = 1, \ldots, c} A' \text{d}y_i \to \Omega_{A'/R} \to 0 $$ where the class of $\pi y_i - g_i$ in $I/I^2$ is mapped to $- \text{d}g_i + \pi \text{d}y_i$ in the next term. Here we have used Algebra, Lemma \ref{algebra-lemma-NL-surjection} to compute $\NL_{A'/A[y_1, \ldots, y_c]}$ and we have used that $R \to A[y_1, \ldots, y_c]$ is smooth, so $H_1(\NL_{A[y_1, \ldots, y_c]/R}) = 0$ and $\Omega_{A[y_1, \ldots, y_c]/R}$ is a finite projective (a fortiori flat) $A[y_1, \ldots, y_c]$-module which is in fact the direct sum of $\Omega_{A/R} \otimes_A A[y_1, \ldots, y_c]$ and a free module with basis $\text{d}y_i$. To finish the proof it suffices to show that $\text{d}g_1, \ldots, \text{d}g_c$ forms part of a basis for the finite free module $\Omega_{A/R, \mathfrak p}$. Namely, this will show $(I/I^2)_\mathfrak p$ is free on $\pi y_i - g_i$, the localization at $\mathfrak p$ of the middle map in the sequence is injective, so $H_1(\NL_{A'/R})_\mathfrak p = 0$, and that the cokernel $\Omega_{A'/R, \mathfrak p}$ is finite free. To do this it suffices to show that the images of $\text{d}g_i$ are $\kappa(\mathfrak p)$-linearly independent in $\Omega_{A/R, \mathfrak p}/\pi = \Omega_{(A/\pi A)/\kappa, \mathfrak p}$ (equality by Algebra, Lemma \ref{algebra-lemma-differentials-base-change}). Since $\kappa \subset \kappa(\mathfrak p) \subset \Lambda/\pi \Lambda$ we see that $\kappa(\mathfrak p)$ is separable over $\kappa$ (Algebra, Definition \ref{algebra-definition-separable-field-extension}). The desired linear independence now follows from Algebra, Lemma \ref{algebra-lemma-computation-differential}.