  We introduce the $N=2$ Lie conformal superalgebras ${\frak {K}}(p)$ of Block
type, and classify their finite irreducible conformal modules for any nonzero
parameter $p$. where $p$ is a nonzero complex number. In particular, we show
that such a conformal module admits a nontrivial extension of a finite
conformal module $M$ over $K_2$ if $p=-1$ and $M$ has rank $(2+2)$, where $K_2$
is an $N=2$ conformal subalgebra of ${\frak {K}}(p)$. As a byproduct, we obtain
the classification of finite irreducible conformal modules over a series of
finite Lie conformal superalgebras ${\frak k}(n)$ for $n\ge1$. Composition
factors of all the involved reducible conformal modules are also determined.
