Is there any known result of decomposing multivariable power series over $p$-adic field into product of single variable power series ?

For example, consider the following power series in $n$ variables:$$ f(x_1,~x_2, \cdots, x_n)=\sum_{j_1,~j_2,\cdots, j_n=0}^{\infty} a_{j_1,~j_2, \cdots, j_n} \prod_{k=1}^{n} (x_k-c_k)^{j_k}.$$

Now we want to express $f(x_1,~x_2, \cdots, x_n)$ in the following way: $$ f(x_1,~x_2, \cdots, x_n)=\left( \sum_{i_1=0}^{\infty} a_{i_1} x_1^{i_1}\right) \cdot \left( \sum_{i_2=0}^{\infty} a_{i_2} x_2^{i_2}\right) \cdots \left( \sum_{i_n=0}^{\infty} a_{i_n} x_n^{i_n}\right).$$ When and under which condition is it possible?

Is there any results or notes or resources available in this regard ?

Thanks,

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