  Let $C_k$ be the cycle of length $k$. For any graph $G$, a subset $D
\subseteq V(G)$ is a $C_k$-isolating set of $G$ if the graph obtained from $G$
by deleting the closed neighbourhood of $D$ contains no $C_k$ as a subgraph.
The $C_k$-isolation number of $G$, denoted by $\iota(G,C_k)$, is the
cardinality of a smallest $C_k$-isolating set of $G$. Borg (2020) and Borg et
al. (2022) proved that if $G \ncong C_3$ is a connected graph of order $n$ and
size $m$, then $\iota(G,C_3) \leq \frac{n}{4}$ and $\iota(G,C_3) \leq
\frac{m+1}{5}$. Very recently, Bartolo, Borg and Scicluna showed that if $G$ is
a connected graph of order $n$ that is not one of the determined nine graphs,
then $\iota(G,C_4) \leq \frac{n}{5}$. In this paper, we prove that if $G \ncong
C_4$ is a connected graph of size $m$, then $\iota(G,C_4) \leq \frac{m+1}{6}$,
and we characterize the graphs that attain the bound. Moreover, we conjecture
that if $G \ncong C_k$ is a connected graph of size $m$, then $\iota(G,C_k)
\leq \frac{m+1}{k+2}$.
