Long live Galileo?

Generically, both $x_B$ and $t_B$ are functions of both $x_A$ and $t_A$ . In Galilean relativity, $x_B=X(t_A,x_A,v)$ while $t_B=t_A$ . However there is no mathematical reason why $t_B$ cannot also be a function of $x_A$ and $v$ . There is also no reason why $x_A=x_B$ or $x_A \neq x_B$ as $v\rightarrow 0$ , as I can shift my origin or not between two coordinate systems/inertial frames.

We now mention an observational fact: the speed of light is experimentally equal to one in every inertial reference frame. Can the following transformation law for $t_B,x_B$ as a function of $t_A,x_A$ be true?

$x_B=x_A-vt_A, t_B=t_A$