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Resistivity and magnetic susceptibility measurements under external pressure were performed on NaFe _1− _x Co _x As ( x  = 0, 0.01, 0.028, 0.075 and 0.109) single crystals. For both underdoped and optimally doped NaFe _1− _x Co _x As, the maximum T _c reached as high as 31 K under certain pressures. Meanwhile the overdoped sample with x  = 0.075 also exhibits a positive pressure effect on T _c , and an enhancement of T _c by 13 K is achieved under a pressure of 2.3 GPa. All of these superconducting samples show large positive pressure coefficients on superconductivity, being distinct from those of Ba(Fe _1− _x Co _x ) _2 As _2 . However, the superconductivity cannot be induced by pressure in heavily overdoped non-superconducting NaFe _0.891 Co _0.109 As. These studies reveal that the electronic structure is very different between superconducting and heavily overdoped non-superconducting NaFe _1− _x Co _x As, consistent with the observation of angle-resolved photoemission spectroscopy.