See the deer in this image? This is from a Japanese card game called hanafuda. The deer (shika) shows up on the card for the tenth (tou) month, and from the we got shikatou (鹿十） which went through a phonetic change to shikato (シカト) over the years.
Now, you already know this if you can also read the description on the image, but shikato means to ignore something, because the deer looks like it is turning it's head away. More accurately, it seems to be used like snub in English.
You may have also caught that this was a "chivalrous" word. Yeah, in Japan, chivalry is something a romantic yakuza would use, so this was once underworld slang. Gogen says it was gamblers' cant. That's not really surprising, as hanafuda have a long history of being used in yakuza gambling dens. The zokugojiten says it spread throughout the delinquent world, until everyone was using it.
Finally, it seems to be a good word to use for being turned down when you are out nampaing, as this YouTube video shows:
So it is neither deer nor fish, but the birds that get away. Unless you are a dog. Wanna 'nother animal in this post? Get this, since the crane (tsuru) on the first (ichi) month's card is also turning away, sometimes tsuruichi is substituted for shikato, but only by someone that is probably making a mistake as a joke.
Usage examples I found and took a stab at glossing:しかとうとぼける [old example] gloss: turn one's head, whistle a little tune, and play dumb.
しかとしよう gloss: Forget about that creep.
しかとかよ gloss: Hey, who do you think you are ignoring?!
しかとすんなよ gloss: Don't you walk away from me!
ツルイチされた! gloss: I had an endearing conversation with the hand.
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