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We prove that for every countable set $\\Gamma$ with $\\vert \\Gamma \\vert \\geq 2$, the Banach space $\\bigoplus_{\\gamma \\in \\Gamma}^{c_0} X_\\gamma $ satisfies the Mazur--Ulam property, whenever the Banach space $X_\\gamma $ is strictly convex with dim$((X_\\gamma )_{\\mathbb{R}})\\geq 2$ for every $\\gamma $. 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We prove that for every countable set $\\Gamma$ with $\\vert \\Gamma \\vert \\geq 2$, the Banach space $\\bigoplus_{\\gamma \\in \\Gamma}^{c_0} X_\\gamma $ satisfies the Mazur--Ulam property, whenever the Banach space $X_\\gamma $ is strictly convex with dim$((X_\\gamma )_{\\mathbb{R}})\\geq 2$ for every $\\gamma $. 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