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Abstract We consider a product $$X=E_1\times \cdots \times E_d$$ of elliptic curves over a finite extension $$K$$ of $${\mathbb{Q}}_p$$ with a combination of good or split multiplicative reduction. We assume that at most one of the elliptic curves has supersingular reduction. Under these assumptions, we prove that the Albanese kernel of $$X$$ is the direct sum of a finite group and a divisible group, extending work by Raskind and Spiess to cases that include supersingular phenomena. Our method involves studying the kernel of the cycle map $$CH_0(X)/p^n\rightarrow H^{2d}_{\acute{\textrm{e}}\textrm{t}}(X, \mu _{p^n}^{\otimes d})$$. We give specific criteria that guarantee this map is injective for every $$n\geq 1$$. When all curves have good ordinary reduction, we show that it suffices to extend to a specific finite extension $$L$$ of $$K$$ for these criteria to be satisfied. This extends previous work by Yamazaki and Hiranouchi.