= Solution
Take the <electric dipole moment> along the polar axis, $\mathbf p=p\hat{\mathbf z}$. The <electric field> is minus the gradient of the <electric potential>; in spherical polar coordinates,
$$
E_r=-\partial_r\phi=\frac{2p\cos\theta}{4\pi\epsilon_0r^3},\qquad
E_\theta=-\frac1r\partial_\theta\phi=\frac{p\sin\theta}{4\pi\epsilon_0r^3},\qquad E_\varphi=0.
$$
Since $\mathbf p=p(\cos\theta\,\hat{\mathbf e}_r-\sin\theta\,\hat{\mathbf e}_\theta)$ and $\mathbf p\cdot\hat{\mathbf e}_r=p\cos\theta$, the radial and angular components combine as
$$
\boxed{\mathbf E(\mathbf r)=\frac{3(\mathbf p\cdot\hat{\mathbf e}_r)\hat{\mathbf e}_r-\mathbf p}{4\pi\epsilon_0r^3},\qquad r>0.}
$$
The origin is excluded because the point-dipole potential is singular there.
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