Let $B$ be the abelian category of 3-term sequences of vector spaces and morphisms $V^{(1)}\to V^{(2)}\to V^{(3)}$ (the composition can be nonzero).  There are 6 indecomposable objects in this category; denote them by $E_1$, $E_2$, $E_3$, $E_{12}$, $E_{23}$, and $E_{123}$.  Here $dim E^{(i)}_J=1$ for $i\in J$ and $0$ otherwise.  Let $A\subset B$ be the full additive subcategory whose objects are the direct sums of all the indecomposables except $E_{12}$ and $f$ be the morphism $E_3\to E_{123}$.

Then one has $Coker_B f=E_{12}$, $Coker_A f=E_1$ and $Im_A f=E_{23}$, while $Coim_A f=E_3$ and therefore $Coker_A(Coim_A f\to Im_A f)=E_2\ne0$.  It is easy to see that any morphism in $A$ with zero kernel and cokernel has also zero kernel and cokernel in $B$, hence is an isomorphism.

Further discussion can be found in my paper http://arxiv.org/abs/1006.4343 , Example A.5(7) (pages 59-60 in version 2).