**EDIT.** New version, addressing questions in the comments.

(1) Affine and smooth implies what you want. 

Indeed, suppose $\mathcal{X}$ is smooth and that $H^1(X_0, T_{X_0/\mathbb{C}}) = 0$ where $X_0/\mathbb{C}$ is the special fiber and where $T_{X_0/\mathbb{C}}$ is the tangent bundle. This is of course satisfied if $X_0$ is affine. 

I claim that in this case $\mathcal{X}$ is actually constant, i.e. $\mathcal{X}$ is the $t$-adic completion of $X_0 \times {\rm Spec}\, \mathbb{C}[[t]]$; if this is true then the constant family $X_0 \times (\text{unit disc})$ is the desired extension. 

The claim follows by usual deformation theory: one shows by induction on $n$ that $X_n = \mathcal{X}\otimes_{\mathbb{C}[[t]]} \mathbb{C}[t]/(t^{n+1})$ is isomorphic over $\mathbb{C}[[t]]$ to $X_0 \otimes_\mathbb{C} \mathbb{C}[t]/(t^{n+1})$. For the induction step, basic deformation theory (e.g. Fantechi's and Illusie's articles in "FGA explained") tells you that liftings of $X_n \cong X_0 \otimes_\mathbb{C} \mathbb{C}[t]/(t^{n+1})$ to $\mathbb{C}[t]/(t^{n+2})$ are a torsor under $H^1(X_0, T_{X_0/\mathbb{C}})$. Since this group vanishes, there is only one such lifting, up to a non-canonical isomorphism. In the limit, we obtain the desired isomorphism.

See also the results of Renee Elkik *Solutions d’équations à coefficients dans un anneau hensélien* Annales scientifiques de l’É.N.S. 4e série, tome 6, no 4 (1973), p. 553-603. She shows (see Theorem 6, p. 580) that a smooth algebra over a ring $R_0= R/I$ can always be lifted (not only formally) over  $R$ if $R$ is noetherian and henselian along $I$.

(2) Smooth affines really are the only case when the answer is positive.

Indeed, take an elliptic curve over $\mathbb{C}[[t]]$ with divergent $j$-invariant, for example
$$ E\colon y^2 z = x(x-z)(x-\lambda z), \quad \lambda = \sum_{n\geq 0} n! t^n. $$ 
Then the $j$-invariant 
$$ j(\lambda) = 258 \frac{(1-\lambda(1-\lambda))^3}{(\lambda(1-\lambda))^2} \in \mathbb{C}[[t]] $$
is likely not convergent. If $E$ was the completion of an analytic family (I am identifying $E$ with the corresponding formal scheme which does not make much difference since $E$ is proper), the $j$-invariant would have to have positive radius of convergence.

The same example works in the affine but singular case by taking the affine cone, i.e. 
$$ \mathcal{X} = {\rm Spf}\, \mathbb{C}[[t]]\{x,y,z\} / (y^2 z - x(x-z)(x-\lambda z), $$
or the same with $[[x,y,z]]$.

Perhaps the simplest example is the plane curve singularity with four lines meeting at a point whose cross-ratio $\lambda$ is divergent, i.e.
$$ xy(x+y)(x-\lambda y).$$