Contradiction of general rules of bond character
Although atomic bond in a compound such as $\ce{M+X-}$ (e.g., $\ce{Na+Cl-}$) is considered to be 100% ionic, in reality, it also has some covalent character. An explanation for the partial covalent character of an ionic bond has been given by Kazimierz Fajans in 1923 (Ref.1). According to Fajans, if two oppositely charged ions (say $\ce{M+}$ and $\ce{X-}$) are brought together, the nature of the bond between them depends upon the effect of one ion on the other. In inorganic chemistry, these Fajans' rules are used to predict whether a chemical bond will be covalent or ionic, and depend on the charge on the cation and the relative sizes of the cation and anion, summarized in the following table:
$$
\begin{array}{c|c}
\hline
\bf{\text{Ionic}} &amp; \bf{\text{Covalent}} \\
\hline
\text{Low positive charge }(\text{e.g., } \ce{Na+}) &amp; \text{High positive charge }(\text{e.g., } \ce{Al^3+})  \\
\text{Large cation }(\text{e.g., } \ce{K+}) &amp; \text{Small cation }(\text{e.g., } \ce{Li+})  \\
\text{Small anion }(\text{e.g., } \ce{F-}) &amp; \text{Large anion }(\text{e.g., } \ce{I-})\\  
\hline
\end{array}
$$
When the two oppositely charged ions, $\ce{M+}$ and $\ce{X-}$, approach each other, the positive ion ($\ce{M+}$) attracts electrons on the outermost shell of the anion ($\ce{X-}$) and repels its ($\ce{X-}$) positively charged nucleus. This results in the distortion, deformation, or polarization of the anion. If the polarization is quite small, the forming bond between them would be more ionic, while that would be more covalent, if the degree of polarization is large.
The power of an ion (cation) to distort the other ion is known as its polarization power and the tendency of an ion (anion) to get polarized by the other ion is known as its polarisability. Based on the (cation’s) polarizing power or (anion’s) polarisability, the formation of covalent bond is favored by the following factors:

Small positive ion (Cation): The smaller the size of cation, the greater is its polarising power. Due to greater concentration of positive charge on a small area, the smaller cation has high polarising power. Thus, the greater will be the covalent nature of its bond with an anion. This explains why $\ce{LiCl}$ is more covalent than $\ce{KCl}$.
Large negative ion (Anion): The larger the size of anion, the greater is its polarizability, i.e. susceptibility to get polarized. It is due to the fact that the outer electrons of a large anion are loosely held and hence can be more easily pulled out by the cation. Thus the greater will be the covalent nature of the bond with a cation. This explains why iodides (e.g., $\ce{Kl}$), among halides (e.g., $\ce{KF}$ and $\ce{KCl}$), are most covalent in nature.
Large charge on either of the two ions: As the charge on an ion increases, the electrostatic attraction of the cation for the outer electrons of the anion also increases. As a result, the cation’s ability for forming the covalent bond increases (see the similarity with second point). For example, the covalency increases in the chlorides of $\ce{Na+, Mg^2+,}$ and $\ce{Al^3+}$ in the order of $\ce{NaCl &lt; MgCl2 &lt; AlCl3}$.
Electronic configuration of the cation:  For the two ions of the same size and charge, one with a pseudo-noble gas configuration (i.e. 18 electrons in outer-most shell) than a cation with noble gas configuration (i.e. 8 electrons in outermost shell) will be more polarizing. Thus copper(I) chloride is more covalent than sodium chloride although $\ce{Cu+}$ ion ($\pu{1.04 \mathring A}$) and $\ce{Na+}$ ion ($\pu{1.05 \mathring A}$) have similar size and charge.

References:

Kazimierz Fajans, “Struktur und Deformation der Elektronenhüllen in ihrer Bedeutung für die chemischen und optischen Eigenschaften anorganischer Verbindungen,” Naturwissenschaften 1923, 11, 165–172 (https://doi.org/10.1007/BF01552365).
Noam Agmon, “Isoelectronic Theory for Cationic Radii,” J. Am. Chem. Soc. 2017, 139(42), 15068–15073 (https://doi.org/10.1021/jacs.7b07882).