Do you want the IC to contain everything? At 44kHz your device will need several MB of storage for each minute of audio. The storage can be ram, CD, hard disk, solid state hard disk.
For playback you might use similar circuitry that goes into a CD player, or a modern note-taking recorder which has no moving parts.
To record, an ADC converter is the middleman component. Your circuitry must take its data output, and fill memory at 44k samples per second.
The commercial devices don't aim for 44kHz speed. I imagine they realize it's overkill in a standalone device.