Sources

Askok 1472 Inventory of Lamont Papers, 51
?Askaig 1519 ER xiv, p. 626
Askak 1541 Inveryne Charters (From Lamont Papers, 158)
Askog 1552-3 Lamont Papers, 191
Ascoks 1564 Retours (Bute) 6
Nedderaskog 1628 Retours (Bute) 39
Askoge 1618 Inventory of Lamont Papers, 430
Askoge 1619 Inventory of Lamont Papers, 433
Nether Ascog 1637 Retours (Bute) 49
Askock, Ascock 1654 Blaeu map Bute
Ascog 1801 Langlands
Ascoghouse 1820 Thomson

There is also Asgog with Loch and Castle on mainland at NR6467:
Askak 1472 RMS ii 1059
Askak 1538-9 RMS iii, 1882
Loch of Ascog 1637 Retours (Bute) 49
Ascok 1654 Blaeu map Bute
L. Ascaig 1747-55 Roy
Arsock 1747-55 Roy
Castle Ascaig 1747-55 Roy
Ascoig 1801 Langlands
Ascoig 1820 Thomson

The derivation of this name is obscure, but the first element is presumably ON, askr, ‘ash’ . The seond element is not ON vík, but something presumably ending containing –agg- or –ogg-. If the two Asgogs above are of identical derivation, then it may be that the final element is –ag in Gaelic, slenderised in the genitive to –aig (as in Loch and Castle Ascaig).