Here's the add for the race (see the third Column), as it appeared in the December 24, 1886 edition of The Black Range:
What about the Weather?
We've had several questions about the weather in Fairview on Christmas Day, 1886.
Try looking at the Historical Climate Information page of the Western Regional Climate Center.
Found something else?
Have you found any more information? Submit your link here. If it looks good, we'll add it!
It was a tie!
There is an image on flicker of the December 31st issue of the Black Range indicating the race was a tie, with a winner to be determined at a run off on January 10th!
Look in the second column, third paragraph of the text!
This has been confirmed by Fray Angélico Chávez History Library who have posted their scan on Facebook!
We have our answer!
The Jan 14 article
Here is the page of the Friday, January 14th edition of the Black Range newspaper. According to the New Mexico Newspaper Project, it started as a weekly paper in 1882. According to the masthead, it was published "every Friday."
Were there issues between the December 24th edition, and the one from January 14th? Maybe. The December 24th issue indicates it is volume V #38. The January 14th issue says it is V 41, suggesting two issues that have been lost: Dec 31st, and Jan 7th.
What we know
The ad promises a race between Dude and Perico, in Fairview (now known as Winston, NM) with each side putting in $500 for $1000 grand prize. The Bureau of Labor Statistics' inflation calculator indicates that $1000 in 1913 (the earliest year for which data is available) would be worth over $20,000 today. So, this is not an insubstantial prize.
The race reported in the January 14th article notes the same prize between the same horses, with Perico winning by six inches. However, this article reports the race having taken place in Chloride, which is almost 3 miles away.
Furthermore, the article says the race took place last Monday, suggesting a January 10th race. The article doesn't mention the earlier race, either to say "Perico wins again" nor to indicate that the race had been a tie before. On the other hand, it doesn't mention that the race was rescheduled either. Any of these might have been covered in the missing editions of the newspaper, if they existed. Unfortunately, our library didn't open until 1901.