When we have only one normal direction the picture is somewhat simpler than in the general case.
Introduce $h_i{}^\beta{}_\alpha$ by $ \nabla^{\perp}_{e_i}e_{\alpha} = h_i{}^\beta{}_\alpha e_\beta $ (Einstein summation assumed), $i=1,\dots,n$, $\alpha, \beta = n+1,\dots,n+p$, as it is assumed in the question.
Claim. $h_i{}^\beta{}_\alpha = - h_i{}^\alpha{}_\beta$.
Proof. $0 = \nabla_{e_i}\langle e_\alpha , e_\beta \rangle = \langle \nabla^{\perp}_{e_i}e_{\alpha} , e_\beta \rangle + \langle e_\alpha , \nabla^{\perp}_{e_i}e_{\beta} \rangle = h_i{}_\beta{}_\alpha + h_i{}_\alpha{}_\beta$.
Corollary. When $p=1$ (i.e. in the case of hypersurfaces) $ \nabla_{e_i}^{\perp} e_\alpha=0 $
Proof. The only skew-symmetric $1\times1$-matrix is $(0)$.