On a continued fraction algorithm in finite extensions of ${\text{Q}}_p$ and its metrical theory

We develop a continued fraction algorithm in finite extensions of ${\text{Q}}_p$ generalising certain algorithms in ${\text{Q}}_p$, and prove the finiteness property for certain small degree extensions. We also discuss the metrical properties of the associated continued fraction maps for our algorithms using subsequence ergodic theory and moving averages.