  A $p$-adic Schr\"{o}dinger-type operator $D^{\alpha}+V_Y$ is studied.
$D^{\alpha}$ ($\alpha>0$) is the operator of fractional differentiation and
$V_Y=\sum_{i,j=1}^nb_{ij}<\delta_{x_j}, \cdot>\delta_{x_i}$
$(b_{ij}\in\mathbb{C})$ is a singular potential containing the Dirac delta
functions $\delta_{x}$ concentrated on points $\{x_1,...,x_n\}$ of the field of
$p$-adic numbers $\mathbb{Q}_p$. It is shown that such a problem is well-posed
for $\alpha>1/2$ and the singular perturbation $V_Y$ is form-bounded for
$\alpha>1$. In the latter case, the spectral analysis of $\eta$-self-adjoint
operator realizations of $D^{\alpha}+V_Y$ in $L_2(\mathbb{Q}_p)$ is carried
out.
