The three synoptics (that is: Matthew, Mark, and Luke) all tell us that, during his final journey to Jerusalem, Jesus healed a blind man (or two) near Jericho. Matthew and Mark both tell us that this occurred after he left Jericho. This is Mark 10:46:
Matthew 20:29-30 is similar. On the other hand, this is Luke 18:35:
For the avoidance of doubt, Luke is clearly talking about the same incident as Mark, because in both cases, the man shouts out "Jesus, Son of David, have mercy on me!" and in both cases, the crowd was irritated. (Compare Mark and Luke.) Matthew's account also contains these details, though with two blind men rather than one. So, did this happen as Jesus left Jericho, or as he approached it?
Looking Unto Jesus argues that our English translations are wrong, and that the verse in Luke is only saying that Jesus was near Jericho, which could mean he had just left it. However, this response ignores what Jesus does immediately after healing the blind man. This is Luke 19:1:
So Luke's account puts the healing of the blind man before, not after Jesus "entered Jericho".
The NIV Study Bible suggests that there were two different cities called "Jericho", Old Jericho (largely abandoned) and New Jericho, and that Luke was thus talking about a different "Jericho" from Mark and Matthew. I don't find this convincing. None of these writers makes any effort to distinguish between two different cities; they all seem to assume that there's one place that is unambiguously referred to as "Jericho".
But this is probably the strongest reply open to an infallibilist.
Luke used Mark's gospel as a source, so why does his order not agree? I can only speculate about this, but I myself misread Mark when I first looked at it. Mark's verse contains the word "Jericho" twice, and my eye jumped from one to the other, so that I thought I had read something like:
It's just speculation, but perhaps something similar happened as Luke read the passage in Mark.