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In Theorem 14.37 of Variational Analysis by Rockafellar and Wets, it shows that for any normal integrand $f:T\times\mathbb{R}^n\to\overline{\mathbb{R}}$, the function $p:T\to\overline{\mathbb{R}}$ given by $$p(t):=\inf~f(t,\cdot)$$ is measurable, and the mapping $P:T\rightrightarrows\overline{\mathbb{R}}$ given by $$P(t):=\arg\min~f(t,\cdot)$$ is closed-valued and measurable.

According to the authors, many results in the book may be extended to spaces other than Euclidean. I am wondering is there a standard reference for the result if we replace $\mathbb{R}^n$ by a Polish (complete and separable metric) space?

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I don't know any reference for the exact result you are looking for, but all the ingredients used in the proof by Rockafellar and Wets apply also to Polish spaces. You can find the relevant material in chapter 18 of Infinite Dimensional Analysis by Aliprantis and Border, 2006, as well as chapters III and VII (only the beginning of the latter chapter matters) in Convex Analysis and Measurable Multifunctions by Castaing and Valadier, 1977.

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