  We develop an analytical method to prove congruences of the type $$
\sum_{k=0}^{(p^r-1)/d}A_kz^k \equiv
\omega(z)\sum_{k=0}^{(p^{r-1}-1)/d}A_kz^{pk} \pmod{p^{mr}\mathbb Z_p[[z]]}
\quad \text{for}\; r=1,2,\dots, $$ for primes $p>2$ and fixed integers
$m,d\ge1$, where $f(z)=\sum_{k=0}^\infty A_kz^k$ is an "arithmetic"
hypergeometric series. Such congruences for $m=d=1$ were introduced by Dwork in
1969 as a tool for $p$-adic analytical continuation of $f(z)$. Our proofs of
several Dwork-type congruences corresponding to $m\ge2$ (in other words,
supercongruences) are based on constructing and proving their suitable
$q$-analogues, which in turn have their own right for existence and potential
for a $q$-deformation of modular forms and of cohomology groups of algebraic
varieties. Our method follows the principles of creative microscoping
introduced by us to tackle $r=1$ instances of such congruences; it is the first
method capable of establishing the supercongruences of this type for general
$r$.
