Let $\mathbf{Q}_p \langle X_1,\dots,X_n \rangle$ be the $n$-variable Tate algebra, i.e. the subalgebra of $\mathbf{Q}_p[[X_1,\dots ,X_n]]$ of power series which converge on the closed unit polydisk in $\mathbf{C}_p^n$. Is $\mathbf{Q}_p \langle X_1, X_2 \rangle$ projective as a $\mathbf{Q}_p \langle X_1 \rangle$-module? Note that $\mathbf{Q}_p \langle X_1 \rangle$ is a regular ring of dimension one.

I am well aware that this is the "wrong question", since it's a mild (severe?) sin to forget the topological structures of the objects in question, but the answer is "yes" in a suitable category of Banach $\mathbf{Q}_p \langle X_1 \rangle$-modules, so I wonder...