We show the existence of uniformly locally attractive solutions for a nonlinear Volterra integral equation of convolution type with a general kernel. We use methods and techniques of fixed point theorems and properties of measure of noncompactness. We extend earlier results obtained in the context of integral equations of fractional order. We give new insights about a new and striking relation between the size of data and the fractional order $\alpha >0$ of the kernel $k(t)=t^{\alpha -1}/\Gamma (\alpha )$.