This is true for any variety $X$ (i.e. a scheme of finite type over a field).
Claim 1: Let $X$ be any variety with irreducible components $X_i$. Let $\varphi: E\rightarrow E$ be any $X$-automorphism of a vector bundle $E$ over $X$. Let $\varphi_i$ be the restriction of $\varphi$ to $\mathbb{P}(E|_{X_i})$. If $\mathbb{P}(\varphi_i)^*$ is the identity for all $i$, then $\mathbb{P}(\varphi)^*$ is the identity.
Proof. Let $\rho_i:\mathbb{P}(E|_{X_i})\rightarrow \mathbb{P}(E)$ be the inclusion. Then $\rho_{i*}\circ \mathbb{P}(\varphi_i)^*= \mathbb{P}(\varphi^*)\circ \rho_{i*}$ by Proposition 1.7 of Fulton. The result follows since the images of $\rho_{i*}$ generate $\mathrm{CH}(\mathbb{P}(E))$. $\square$
Claim 2: Let $X$ be any variety and let $\varphi:E\rightarrow E$ be any $X$-automorphism of a vector bundle $E$ over $X$. Then $\mathbb{P}(\varphi)^*$ is the identity.
Proof. We'll proceed by Noetherian induction on the dimension of the irreducible components of $X$. For this purpose, let $X$ be any variety with irreducible components $X_i$ having dimension $\max_i\{ \mathrm{dim}(X_i)\}\leq n$.
For the base case, we observe that if $X$ is a point, then any automorphism of $\mathbb{P}^m_X$ has to take $\mathcal{O}(1)$ to itself in $\mathrm{Pic}(X)$ and this generates the ring $\mathrm{CH}^*(\mathbb{P}^m_X)$.
Assume that the result is true for all $X$ with irreducible components $X_i$ with $\max_i\{\mathrm{dim}(X_i)\}\leq n-1$. By Claim 1, it suffices to show that this implies the result when $X$ is irreducible. Then for any open $U\subset X$ there is a commuting ladder with exact rows
$$\begin{matrix} \mathrm{CH}(\mathbb{P}(E|_Z))&\rightarrow& \mathrm{CH}(\mathbb{P}(E))&\rightarrow& \mathrm{CH}(\mathbb{P}(E|_U))&\rightarrow & 0 \\ \downarrow && \downarrow && \downarrow\\ \mathrm{CH}(\mathbb{P}(E|_Z))&\rightarrow& \mathrm{CH}(\mathbb{P}(E))&\rightarrow& \mathrm{CH}(\mathbb{P}(E|_U))&\rightarrow & 0\end{matrix}$$ where the vertical morphisms are induced by $\mathbb{P}(\varphi)^*$. By our induction hypothesis, the left-most arrow is the identity whenever $U$ is nonempty. Taking a limit over the inductive system of closed subschemes $Z\subset X$, we get a commutative ladder with exact rows given by localization $$\begin{matrix} \varinjlim_Z \mathrm{CH}(\mathbb{P}(E|_Z))&\rightarrow& \mathrm{CH}(\mathbb{P}(E))&\rightarrow& \mathrm{CH}(\mathbb{P}(E|_{\eta}))&\rightarrow & 0 \\ \downarrow && \downarrow && \downarrow\\ \varinjlim_Z \mathrm{CH}(\mathbb{P}(E|_Z))&\rightarrow& \mathrm{CH}(\mathbb{P}(E))&\rightarrow& \mathrm{CH}(\mathbb{P}(E|_{\eta}))&\rightarrow & 0\end{matrix}$$ where $\eta$ is the generic point of $X$. By our induction, the rightmost vertical arrow is the identity. The groups $\mathrm{CH}(\mathbb{P}(E|_\eta))$ are free so these sequences are (noncanonically) split. A diagram chase finishes the proof. $\square$