Quantum memories are a key building block for optical quantum computers and quantum networks. Rare-earth ion-doped crystals are a promising material to achieve quantum memory using an atomic frequency comb protocol. However, current atomic frequency comb memories typically use bulk materials or waveguides with large cross sections or rely on fabrication techniques not easily adaptable to wafer scale processing. Here, we demonstrate a compact chip-integrated atomic frequency comb in rare-earth-doped thin-film lithium niobate. Our optical memory exhibits a broad storage bandwidth exceeding 100 MHz and optical storage time as long as 250 ns. The enhanced optical confinement in this device leads to three orders of magnitude reduction in optical power required for a coherent control as compared to ion-diffused waveguides. These compact atomic frequency comb memories pave the way toward scalable, highly efficient, electro-optically tunable quantum photonic systems that can store and manipulate light on a compact chip.