Ice pips and Lava Eels, the latter before reaching Volcano, are among the hardest catches or rather, encounters, in the entire game. Oh and octopi, damn that octopi! I spent entire days (in-game time), in order to finally meet one. Which means, abandoning everything first thing in the morning and going to mines, to fish.
You want to have that tackle equipped on your rod, which makes the fishing bar deplete slower, when the fish hops around to escape. I still cannot get over the traumatic experience of my first ice-pip bite, going to waste, after having spent so much time hunting for it. You might also want to consume a fishing-buffing food in advance and utilize some bait, in tandem.