Gift aid for non taxpayers?

Income for tax credit purposes is reduced by any money donated to charity by gift aid, potentially gaining 41p per £ donated. Does this apply to a non taxpayer too, who is unable to gift aid?

If not, would the "solution" be for the non taxpayer to "gift-aid" the nett amount, and repay the tax recovered to HMRC? Otherwise non-taxpayers seem unable to take advantage of their charitable givings for tax credit purposes