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$$A = \mathbb{F}_p[x_1,\ldots,x_6]/(x_1^p,\cdots, x_6^p, x_1x_2 + x_3x_4 + x_5x_6)$$
does not lift to a finite flat $\mathbb{Z}_p$ algebra. (I am still working out the details of why this does not lift.) This is exampe 3.2(4) of Berthelot-Ogus.