Smart attack details
Given some curve $E/\mathbb F_p$ with order $p$, lift it to $E_0=E/\mathbb Q_p$ and define the subgroups of $E/\mathbb Q_p$: $E_r=\{(x,y)\in E/\mathbb Q_p|v_p(x)\leq-2r,v_p(y)\leq-3r\}\cup\{\infty\}$. We have the following isomorphism of groups: $\frac{E_0}{E_1}\cong E/\mathbb F_p$ and $\frac{E_1}{E_2}\cong\mathbb F^+_p$, which the first isomorphism given by reduction mod $p$ last isomorphism given by $\psi:(x,y)\to -\frac x{py}\pmod p$
Assume that $kP=Q$ in $E/\mathbb F_p$, typically if we are only given $P,Q$, findiing $k$ is computationally hard. However since $E/\mathbb F_p$ is anomalous, Smart attack gives us a simple solution:
After lifting the curve to $E/\mathbb Q_p$, let $\tilde P,\tilde Q$ be the lifted points of $P,Q$ respectively, then we have $p\tilde P,p\tilde Q,k\tilde P-\tilde Q\in E_1$ since $E/\mathbb F_p$ is of order $p$. Furthermore, since $k\psi(p\tilde P)-\psi(p\tilde Q)=p\psi(k\tilde P-\tilde Q)=0$, we get that $k=\frac{\psi(p\tilde Q)}{\psi(p\tilde P)}\pmod p$, which is pretty simple to compute.
When the attack fails
The proof fails if $pP\in E_2$, since we get $k=\frac00$. This is noted in the Smart attack's paper, where he notes that for 'canonical lifts', such points exist, and 'canonical lifts' occur at a frequency of $\frac1p$. An example of such a lift can be found at the crypto SE question why the Smart attack fails for a particular lift.
My question is why does hitting the canonical lift cause the attack to fail(which was answered in the comments), and why does it happen with a $\frac1p$ probability?
Some examples of 'canonical lifts':
The curve, $y^2=x^3+3x+5$ over $\mathbb F_7$, has the following 'canonical lifts' when lifted to $\mathbb Q_7$, for $m,n>0,m,n\in\mathbb Z$ and $p=7$:
$$y^2=x^3+\left(3+0p+mp^2\right)x+\left(5+3p+np^2\right)$$ $$y^2=x^3+\left(3+1p+mp^2\right)x+\left(5+2p+np^2\right)$$ $$y^2=x^3+\left(3+2p+mp^2\right)x+\left(5+1p+np^2\right)$$ $$y^2=x^3+\left(3+3p+mp^2\right)x+\left(5+0p+np^2\right)$$ $$y^2=x^3+\left(3+4p+mp^2\right)x+\left(5+6p+np^2\right)$$ $$y^2=x^3+\left(3+5p+mp^2\right)x+\left(5+5p+np^2\right)$$ $$y^2=x^3+\left(3+6p+mp^2\right)x+\left(5+4p+np^2\right)$$
In general(as far as I can tell by direct computation), one finds $p$ 'canonical lifts' mod $p^2$ for any anomalous curves, which satisfies the $\frac1p$ probability that Smart gave.
About the 'canonical lift'(from comments)
If the Frobenius endomorphism $f\in\text{End}\left(E/\overline{\mathbb F_p}\right)$ lifts to a endomorphism $\tilde f\in\text{End}\left(E/\overline{\mathbb Q_p}\right)$, then this is the 'canonical lift' that Smart is referring to, which is less strict than requiring $\text{End}\left(E/\overline{\mathbb F_p}\right)\cong\text{End}\left(E/\overline{\mathbb Q_p}\right)$. Since $\left|E/\mathbb F_p\right|=p$, the dual of $f$, $f^*$ has kernel $E/\mathbb F_p$ and is separable, which implies the dual of $\tilde f$, $\tilde f^*$, has a kernel of order $p$, which is precisely the $p$-torsion elements of $E/\mathbb Q_p$, and is the torsion group $\left(E/\mathbb Q_p\right)_\text{tors}$ of this curve, hence $E_0\cong E_1\times\left(E/\mathbb Q_p\right)_\text{tors}$, hence $pE_0=pE_1$ and $p\tilde P\in E_2$, causing the attack to fail.