Let $k$ be a $p$-adic field. It's possible to make sense of the Haar measure $\mu_{\operatorname{Haar}}$ on $k^n$ as a $k$-valued measure and define integrals
$$\int\limits_{k^n} f(x_1, ... , x_n) d\mu_{\operatorname{Haar}}(x)$$
for suitable analytic $k$-valued functions $f$ on $k^n$. If $\varphi: U \rightarrow V$ is a $k$-analytic isomorphism between open sets $U$ and $V$ in $k^n$, then for suitable complex valued functions $f: k^n \rightarrow \mathbb C$, we have a change of variables formula
$$\int\limits_V f(x)dx_1 \cdots dx_n = \int\limits_U f \circ \varphi(x) |D_{\varphi}(x)|dx_1 \cdots dx_n$$ where $D_{\varphi}$ is the determinant of the Jacobian of $\varphi$, and $dx_1 \cdots dx_n$ is the (real valued) Haar measure on $k^n$. Is there an analogue of this formula for $k$-valued measures and $k$-analytic maps $f: k^n \rightarrow k$? Something like
$$\int\limits_V f(x) d\mu_{\operatorname{Haar}}(x) = \int\limits_U f \circ \varphi(x) D_{\varphi}(x) d\mu_{\operatorname{Haar}}(x)$$ (without the absolute value)?