  For a fixed prime $p$, let $\mathbb C_p$ denote the complex $p$-adic numbers.
For polynomials $A,B\in \mathbb C_p[x]$ we consider decompositions
$A(x)f^2(x)+B(x)g^2(x)=1$ of entire functions $f,\,g$ on $\mathbb C_p$ and try
to improve an impossibility result due to A. Boutabaa concerning transcendental
$f,g$. We also provide an independent proof of a $p$-adic diophantic statement
due to D. N. Clark, which is an important ingredient of Boutabaa's method.
