Telomere length (TL) and/or its rate of change are popular biomarkers of senescence, as telomere dynamics are linked with survival and lifespan. However, the evolutionary potential of telomere dynamics has received mixed support in natural populations. To better understand how telomere dynamics evolve, it is necessary to quantify genetic variation in TL and how such variation changes with age. Here, we analyzed 2,083 longitudinal samples from 1,225 individuals across 16 years, collected from a wild, insular house sparrow () population with complete life history and genetic relatedness data. Using a series of "animal" models, we confirmed that TL changes with age, reflecting senescence in this population. We found TL to be repeatable (14.0%, 95% CrI: 9.1%-19.9%) and heritable (12.3%, 95% CrI: 7.5%-18.2%); and detected a genotype-by-age interaction, meaning that genotypes differ in their rate of change of TL, and additive genetic variance increases at older ages. Our findings provide empirical evidence from a wild population that supports hypotheses explaining the evolution of senescence and highlight the importance of telomere dynamics as a key biomarker of body physiology for the evolution of senescence.