No, we don't even have a full Neanderthal genome.

There are claims that an entire Neaderthal genome has been mapped, see doi: 10.1038/nature12886, from 2014.

There are also some low coverage genomes.

Not sure what has been done since 2014.

(Many of us are actually part Neanderthal so talking about this gets fairly odd. The "pre-admixture" populations had maybe 99.7% shared DNA anyway.)