$\newcommand\ep\varepsilon$Take any $\ep\in(0,1/2)$. Let 
$$A:=A_0\cup A_1\cup F,$$
where $A_0:=[0,\ep)\times B$, $A_1:=(1-\ep,1]\times B$, $F:=[0,1]\times\{0\}$, and $B$ is the unit ball (you did not say if your unit ball is closed or open; let us assume that it is closed). 

You have a well-defined function $f$ on $A$ such that $f|_{A_0}=\rho_0$, $f|_{A_1}=\rho_1$, and $f(\cdot,0)=F_0$. 

Your question is then when $f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$. 

For that it is clearly necessary that $\rho_0$ and $\rho_1$ be extendible to smooth functions $\bar\rho_0$ and $\bar\rho_1$ on the closures $\bar A_0$ and $\bar A_1$ of $A_0$ and $A_1$. Then $f$ can be accordingly extended to a uniquely determined function $\bar f$ on the closure $\bar A$ of $A$. 

So, the question then becomes when $\bar f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$. 

By [Whitney's][1] Theorem I, a sufficient condition for this is that $\bar f$ be of class $C^\infty$ on $\bar A$ in the sense of this paper by Whitney (the set $A$ in the mentioned theorem can be any closed subset of a Euclidean space -- it does not have to be a manifold). 

It is clear that this sufficient condition is also necessary. 

Therefore and because the functions $\bar\rho_0$ and $\bar\rho_1$ are smooth, we conclude: 

>$f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$ iff condition (3.2) in the mentioned paper by Whitney holds for the function $\bar f$ and all the points $x^0$ in the set $[\ep,1-\ep]\times\{0\}$ (with respect to the set $\bar A$). 

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The case $\ep>1/2$ is trivial. In the case when $\ep=1/2$, the obvious necessary and sufficient condition is that the values of the function $\bar\rho_0$ and all its partial derivatives on the set $\bar A_0\cap\bar A_1=\{1/2\}\times B$ be the same as the corresponding values of the function $\bar\rho_1$ and all its partial derivatives.  

  [1]: https://www.ams.org/journals/tran/1934-036-01/S0002-9947-1934-1501735-3/S0002-9947-1934-1501735-3.pdf