  Let $p$ be a rational prime, $v_p$ the normalized $p$-adic valuation on
$\mathbb{Z}$, $q>1$ a $p$-power and $A=\mathbb{F}_q[t]$. Let $\wp\in A$ be an
irreducible polynomial and $\mathfrak{n}\in A$ a non-zero element which is
prime to $\wp$. Let $k\geq 2$ and $r\geq 1$ be integers. We denote by
$S_k(\Gamma_1(\mathfrak{n}\wp^r))$ the space of Drinfeld cuspforms of level
$\Gamma_1(\mathfrak{n}\wp^r)$ and weight $k$ for $A$. Let $n\geq 1$ be an
integer and $a\geq 0$ a rational number. Suppose that $\mathfrak{n}\wp$ has a
prime factor of degree one and the generalized eigenspace in
$S_k(\Gamma_1(\mathfrak{n}\wp^r))$ of slope $a$ is one-dimensional. In this
paper, under an assumption that $a$ is sufficiently small, we construct a
family $\{F_{k'}\mid v_p(k'-k)\geq \log_p(p^n+a)\}$ of Hecke eigenforms
$F_{k'}\in S_{k'}(\Gamma_1(\mathfrak{n}\wp^r))$ of slope $a$ such that, for any
$Q\in A$, the Hecke eigenvalues of $F_k$ and $F_{k'}$ at $Q$ are congruent
modulo $\wp^\kappa$ with some $\kappa>p^{v_p(k'-k)}-p^n-a$.
