Electrolysis involving K+ and Li+ ions
$\ce{Li+}$ 's low standard reduction potential value applies to aqueous solutions only. 

$\ce{Li+}$ has very high polarising power due to its small size (and in turn a high hydration energy). This is why $\ce{Li+}$ has a very low reduction potential compared to other alkali metals in aqueous solutions.

When you deal with a solution of molten metals, the idea of hydration energy makes no sense anymore. Therefore, the electrochemical series order for aqueous solutions isn't applicable to the molten solution of alkali metals.

As you go down the alkali metal group, the ionisation energy decreases. The tendency of an atom to give its electron (or get oxidised) increases. Therefore, potassium oxidises more easily than lithium or in other words, potassium has a lesser tendency to get reduced compared to lithium.

As lithium has a higher tendency to get reduced than potassium, lithium gets deposited instead of potassium.