The author considers a linear recurrence \[ x_{n+1}=a_nx_n+b_n,\qquad n\geq 0,\;x_0\in X \] where \((x_n)\) is a sequence in a Banach space \(X\) and \((a_n)\), \((b_n)\) are given sequences of scalars and vectors in \(X\), respectively. Then, a stability result is proved: Suppose that \(\varepsilon>0\), \(| a| >1\) and an arbitrary sequence \((b_n)\) in \(X\) are given. If a sequence \((x_n)\) in \(X\) satisfies the relation \[ \| x_{n+1}-ax_n-b_n\| \leq\varepsilon,\qquad n\geq 0, \] then there exists a unique sequence \((y_n)\) in \(X\) given by the recurrence \[ y_{n+1}=ay_n+b_n,\qquad n\geq 0,\;y_0\in X \] such that \[ \| x_n-y_n\| \leq\frac{\varepsilon}{| a| -1},\qquad n\geq 2. \] In fact, the main result of the paper is even more general, with constants \(a\) and \(\varepsilon\) replaced by sequences \((a_n)\) and \((\varepsilon_n)\) satisfying some additional assumptions.