Let $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ be sequences of $\mathbb Q_p$ such that the function $f:z\in\mathbb Q_p\to\sum_{n\ge0}a_nz^n+b_nz^{n+1}$converges in $\{|z|_p<1\}$. Assume that the series $\sum_{n\ge0}a_n+b_n$ converges in $\mathbb Q_p$. Can the function $f$ be continued in a larger disk in an analytic function?

Thanks in advance for for any hint or answer.