No, the only physical consistent models where the relativity postulate works (i.e., no special reference frames and the laws of physics would be the same in all reference frames moving with constant velocity) are Galilean and Lorentzian relativity. The proof is given by <PERSON> here (section 11.10).
To ensure relativity, the transformation between reference frames must be of the form \begin{align}x' =& A_v(x+vt)\ t' =& A_v\left[t+\frac{x'}{v}\left(1-\frac{1}{A_v^2}\right) \right] \end{align} where $A_v$ could depend on the velocity $v$. Now we define $V_v$ as $$V_v = \frac{v}{\sqrt{1-\frac{1}{A_v^2}}}$$ With this change of variable, the relativistic transformations read \begin{align}x' =& \frac{x+vt}{\sqrt{1+\frac{v^2}{V_v^2}}}\t' =& \frac{t+\frac{vx}{V_v^2}}{\sqrt{1+\frac{v^2}{V_v^2}}} \end{align} which look a lot like the usual <PERSON> transformations, except for the fact that $V_v$ could depend on $v$.
But we need one more requirement: the relativity transformation must form a group: if you transform from reference frame RF1 to RF2, and then from RF2 to RF3, the you can also transform directly from RF1 to RF3. 804
\begin{align}x'' =& \frac{x'+wt'}{\sqrt{1+\frac{w^2}{V_w^2}}} = \frac{1}{\sqrt{1+\frac{v^2}{V_v^2}}}\frac{(x+vt)+w(t+vx/V_v^2)}{\sqrt{1+\frac{w^2}{V_w^2} }}\=& \frac{1}{\sqrt{1+\frac{v^2}{V_v^2}}\sqrt{1+\frac{w^2}{V_w^2} } }\left[x\color{red}{\left(1+\frac{vw}{V_v^2}\right)}+t(v+w)\right]\ t'' =& \frac{t'+\frac{wx'}{V_w^2}}{\sqrt{1+\frac{w^2}{V_w^2}}} = \frac{1}{\sqrt{1+\frac{v^2}{V_v^2}}}\frac{t+\frac{vx}{V_v^2}+\frac{w}{V_w^2}(x+vt) }{\sqrt{1+\frac{w^2}{V_w^2}}}\=& \frac{1}{\sqrt{1+\frac{v^2}{V_v^2}}\sqrt{1+\frac{w^2}{V_w^2} } }\left[t\color{red}{\left(1+\frac{vw}{V_w^2}\right)}+x\left(\frac{v^2}{V_v^2}+\frac{w^2}{V_w^2}\right)\right] \end{align}
See the parts in red? They should be equal in a relativity transformation. This means that $V_v^2 = V_w^2$, and therefore, $V_v^2$ is a constant. Which values can it take?
* $V^2=0$: Transformations between different reference frames are not possible.
* $V^2<0$: The relativity transformations are rotations, and the periodicity produces problems with casuality.
* $V^2=+\infty$: Galilean relativity.
* $V^2$ some finite positive value: Lorentzian relativity, with speed of light $c=\sqrt{V^2}$. 563